math3491/5492: discrete systems and integrabilityfrank/math5492/lectures.pdf · 2010-12-08 ·...

1

MATH3491/5492:

DISCRETE SYSTEMS AND INTEGRABILITY

Frank W NIJHOFF

University of Leeds, School of Mathematics

0

Aknowledgement:These lecture notes are an adaptation of a text which is intended as a monograph by J.Hietarinta, N. Joshi and F.W. Nijhoff on the same subject.

Preface

These Lectures deal with discrete systems, i.e. models of (physical, biological, economic,etc.) phenomena that are mathematically modelled through equations that involve finite(as opposed to infinitesimal) operations.

When modelling continuous phenomena in nature we are often using differential equationsas the mathematical tools to encode the essence of what is going on in the system we arestudying. The differential equations are believed to encapture the main mechanisms behindthe phenomena, based on either microscopic theory or general principles relevant to thesubject. Whatever the source of the model, solving the basic differential equations is thenbelieved to lead to predictions on the behaviour of the system.

In many circumstances, the phenomena of nature are most adequately described notin terms of continuous but in terms of discontinuous steps (for instance many models inbiology, economy, neural processes, decision processes, etc.). In that case we are dealingwith discrete-time systems, which can mathematically be descrided through:

• recurrence relations;

• difference equations;

• dynamical mappings.

In addition to these types of systems we will also consider:

• functional relations

which are of a slightly different character, but will also arise in connection to discrete systems.All of the above will form the main mathematical tools in these Lectures.

To make a comparison with the continuous theory, let us recall the types of differentialequations we know. These fall into two classes:

1. Ordinary differential equations (ODEs), which are of the form:

F ( y(x), y′(x), y′′(x), . . . ;x ) = 0 , (0.0.1)

in which F is some expression (a function of its arguments) and where x is called theindependent variable, whilst y(x) is the dependent variable since it depends on x. Theprimes denote differentiation w.r.t. x:

y′(x) =dy

dx, , y′′(x) =

d2y

dx2, . . .

1

2

and the aim is to try and solve y as a function of x, wherever possible, from (0.0.1).The solution, if we can find it at all, is not unique, and needs further data in order torender it unique. For instance, this can be done by imposing, in addition to the ODEitself, a number of initial data, e.g. by fixing values of y and some of its derivatives ata given value of x, say at x = 0, namely:

y(0) , y′(0) , . . .

2. Partial differential equations (PDEs), in which case we have more than one independentvariable. For instance, if we have two independent variables, say x and t, a PDE takeson the general form:

F ( y, yx, yt, yxx, yxt, ytt, . . . ;x, t ) = 0 , (0.0.2)

where F denotes again some expression, and we have in its argument the functiony(x, t), the dependent variable depending on both independenet variables, and itspartial derivatives

yx =∂y

∂x, yt =

∂y

∂t, yxx =

∂2y

∂x2, yxt =

∂2y

∂x ∂t, ytt =

∂2y

∂t2, . . .

The theory of PDEs is quite different in character from the theory of ODEs. In fact,PDEs may possess many solutions and the determination of auxiliary data is muchmore complicated. These data may comprise initial values as well as boundary values,and a finite number of auxiliary data (to fix the solution) is no longer sufficient.

Let us turn now to the discrete situation. Let us recall that the derivatives, on whichdifferential equations are built, come from the limit:

dy

dx= lim

h→0

y(x+ h)− y(x)h

, (0.0.3)

and that without taking the limit we actually first encounter the difference operation:

∆hy(x) =y(x+ h)− y(x)

h. (0.0.4)

The derivative (0.0.3) and the differential (0.0.4) are, in fact, quite distinct operations: thefirst is a local operation, involving the function at one point x, whilst the difference operation(0.0.4) is inherently nonlocal and involves two points x and x+h at a distance h. By takingthe limit h→ 0, we actually throw away information and thus the difference operator ∆h isin a sense a more general construct. There is good reason to study continuous systems inwhich the derivative plays the main role: the operations are local, instantaneous, involvingsmooth functions, thus apparently most suitable to describe the apparent smoothness andcontinuity that we observe in Nature.

The study of difference equations seem often to come in hindsight: in those cases whereone cannot calculate solutions of differential equations by analytic means, finite-differenceschemes are used to obtain approximate solutions by numerical methods. Here the discreteequations seem to come as an artefact and a tool rather than being fundamental to the

3

phenomena under study. However, in view of eq. (0.0.3) this may be considered as a logicalreversal: since we get the derivative (0.0.3) by the reductive procedure of a limit from thefinite operation (0.0.4) of a difference, shouldn’t we consider the latter as being the morefudamental?

In modern physics, this question gains even more weight. The underlying structure ofsubatomic physics, based on quantum mechanics, leaves no doubt that there is a fundamentalwidth (given in terms of Planck’s constant ~) beyond which a “continuum” is no longerperceptible, and even may make no longer sense. Thus, maybe the continuum aspect ofNature is an illusion, only of validity on the scale of the macroscopic world. Rather thanconsidering the finite-difference to be the approximation, one actually should consider thederivative to be the approximation (or simplification) of a more fundamental set of tools. Afamous citation by A. Einstein1 declares:

“To be sure, it has been pointed out that the introduction of a space-time con-tinuum may be considered as contrary to nature in view of the molecularstructure of everything which happens on a small scale. It is maintained thatperhaps the success of the Heisenberg method points to a purely algebraic de-scription of nature, that is to the elimination of continuum functions fromphysics. Then, however, we must also give up, by principle, the space-timecontinuum. It is not unimaginable that human ingenuity will some day findmethods which will make it possible to proceed along such a path. At thepresent time, however, such a program looks like an attempt to breathe inempty space.”

Why then haven’t we then progressed further in the theory of difference equations? Itis fair to say that in contrast to the theory of (ordinary or partial) differential equations,the theory of difference equations is even today, in the beginning of the 21st century, stillin its infancy. One reason for this discrepancy is that in spite of the simple appearance ofthe basic difference operation (0.0.4) the development of comprehensive theory has provedto be far more difficult. Many more technical difficulties are met in the development ofdifference calculus than in differential calculus, mainly due to the nonlocal nature of thedifference operator. Furthermore, it is much more difficult to classify difference equationsthan is the case for differential equations: there are just too many possibilities of writingdown difference equations, whereas for differential equations one can easily distinguish themby their order, degree, well-posedness, or amenability to boundary and initial value problems,etc.

But there are also historical reasons. Up to the beginning of the 20th century, mathemati-cians considered the theory of differential equations and the theory of difference equationsvery much as two aspects of one and the same endeavor. It is mostly after the two worldwars that a generation of mathematicians working systematically on difference equations(the famous school of G.D. Birkhoff) disappeared.

It is also important to recall the role of physics as an inspiration in the development ofmathematics, from Newton on. At the beginning of 20th century there were two interestingdevelopments in physics, which also caught the interest of mathematicians: general relativity

1A. Einstein, Physics and Reality, 1936. (Reprinted in Essays in Physics, Philosophocal Library, NewYork, 1950)

4

and quantum theory. Both relied on “new” mathematics, and attention was drawn ondeveloping these tools further. It was only in the 1960’s that attention returned on classicaldynamics, on discrete dynamics and the associated chaos on one hand, and on PDE’s andthe associated soliton equations on the other. Major progress was then made on both ofthese areas in the 1970’s and 1980’s so that both have now developed into paradigms withassociated mature theory.

What has fundamentally changed in the last two decades is the surge of interest in inte-grable difference equations (see, e.g, http://www.side-conferences.net). Integrability isstrongly associated with regularity of the behavior, but this regularity is not due to triviality,but rather due to some underlying mathematical structure that restricts the dynamics in anontrivial way. This has already been well established in the theory of soliton equations.The theory of integrable difference equations (both ordinary and partial) has undergone arevolution since the early 1980’s, and has also achieved major developments, although manyproblems are still open. In fact, the various notions of integrability (of which more in thefollowing) have proven to be a very strong guiding principle which has driven the subjectforward. The subjects which it has strongly affected include the following

• ordinary difference equations (O∆Es);

• partial difference equations (P∆Es);

• integrable discrete-time systems (mappings);

• discrete and “difference” geometry;

• linear and nonlinear special functions, orthogonal polynomials;

• representation theory of (infinite-dimensional and quantum) algebras & groups;

• combinatorics and graph theory.

Some of these developments go back to results from the 19th century and even earlier,but it is fair to say that in combining classical results (from Leibniz, Bernoulli, Gauss,Lagrange, etc.), via the turn of the 19/20th century (Poincare, Birkhoff, Norlund, etc.)with the results of the modern age, the study of integrability in discrete systems forms atthe present time the most promising route towards a general theory of difference equationsand discrete systems.

One of the amazing facts is that in the study of integrable difference equations, we willencounter all the above type of equations, but moreover these types turn out to be quiteinterconnected. In fact, we will see that in many examples we can interpret one and the samesystem either as a dynamical map or finite difference equation, or as an analytic differenceequation, or even as functional equation, without damaging the main integrability aspect ofthe system.

Types of Difference Equations (∆Es)

In these Lectures we are going to consider various types of discrete systems. From a generalperspective let us briefly mention the various types of difference equations that we mightencounter.

5

First, as with differential equations, they divide up between ordinary difference equations(O∆Es) and partial difference equations (P∆Es), depending on whether there is one or morethan one independent variable. The independent variable is discrete, but now this can havedifferent interpretations:

a) the independent variable can take on only integer values, so we denote it by n ∈ Z, , andthus the dependent variable y can be denoted by yn;

b) the independent variable x can take on all values (real or perhaps even complex), butin the equation the dependent variable is evaluated at integer shifts of the indepen-dent variable. Thus, the difference equation will involve evaluations of the dependentvariable y(x) at shifts of x by multiples of a fixed quantity h:

y(x) , y(x+ h) , y(x+ 2h) , . . . x, h ∈ R .

Remark: These two cases are quite different in nature and the associated problems arequite distinct. However, case a) can be viewed as specialisation of case b), by setting

yn ≡ y(x0 + nh)

and by fixing x0 as a starting value, we can consider the values of the idependent variableyn as a stroboscope of the second case.

Remark: Before describing various types of difference equations, let us first make theobvious observation that prior to the the difference operator (0.0.4) there is an even morebasic operation, namely that of the (lattice) shift. This is simply the operation of evaluatingthe dependent variable at a shifted value of the independent variable, and it is useful tointroduce an operator associated with this operation:

Thy(x) = y(x+ h) , (0.0.5)

which we will use from time to time. It is easy to see that the difference operator and itshigher orders can be simply expressed in terms of the shifts Th:

∆hy(x) =1

h(Th − id)y(x) ,

∆2hy(x) =

1

h2(T 2h − 2Th + id)y(x) ,

. . . . . .

∆nhy(x) =

(−1)nhn

n∑

j=0

(nj

)(−Th)n−j y(x)

and thus the nth order difference operator acting on y(x) can be expressed in terms of theshifted variables y(x), y(x+ h), . . . , y(x+ nh) . Thus an equation of the form:

F(y(x),∆hy(x),∆2hy(x), . . . ;x) = 0

can be rewritten in the form:

F(y(x), y(x + h), y(x+ 2h), . . . ;x) = 0 ,

6

where the expression F can be straightforwardly be obtained from F by using the abovementioned substitutions.

n− k n− k + 1 n− k + 2 n+ l − 1 n+ l

←− h −→ → x

Taking into account this remark, we can distinguish the following types of discrete systems:

a) Ordinary finite difference equations: by which we mean recurrence relations of the form

F(yn−k, yn−k+1, . . . , yn+l−1, yn+l;n) = 0 , n ∈ Z, (0.0.6)

where k and l are fixed integers (see Figure) . We can think of (0.0.6) as an iterativesystem where we wish to solve yn at discrete points only. Thus, giving initial valuesat a sufficient number of points, generically y0, y1, . . . , yk+l, we can hope to iteratethe equation and calculate step-by-step yk+l+1 and subsequent values. The equationmay depend explicitely on n, in which case the equation is nonautonomous, but ifF depends only on n through the dependent variables then the equation (0.0.6) isautonomous. The order of the equation is generically given by k+ l, i.e. the number ofinitial data required to achieve a well-defined iteration process. We will sometime referto an equation of the form (0.0.6) as an (k + l + 1)-point map (assuming all “points”at which y is evaluated appear in the expression F).

Finite difference equations of this type can also be viewed as a dynamical mappingas follows. Assuming that we can solve yn+l uniquely from (0.0.6), leading to anexpression of the form

yn+l = Fn(yn−k, . . . , yn+l−1) ,

and introducing the k + l-component vector

yn = (yn−k−l, . . . , yn−1)t

we can rewrite (0.0.6) as a system of equations through the dynamical map:

yn 7→ yn = yn+1 = F n(yn) , (0.0.7)

with F denoting the vector-valued function with components Fj (viewed as a functionof the vector yn).

b) “analytic” (ordinary) difference equations: i.e. equations of the type

F (y(x− kh), y(x− (k − 1)h), · · · , f(x+ lh);x) = 0 , (0.0.8)

where now, even though the equation only involves integer shifts (by an increment h) inthe arument of the dependent variable y(x), the independent variable x is meant to bea continuous variable, and we would like to solve (0.0.10) for functions y(x) in someappropriate class of functions. Clearly, there is indeterminacy in this problem: the

7

solutions cannot be determined fully without additional information, and the solutionis determined only up to a shift over h. This implies that the initial data have tobe given over an entire interval on the real line, or alternatively that the role of“integration constants” is played by periodic functions obeying:

π(x + h) = π(x) .

c) functional equations: This is a different class of equations altogether,taking the form

F (f(x), g(x), f(x + y), g(x+ y), f(x+ y + z), . . . ;x, y, z, . . . ) = 0 , (0.0.9)

which is supposed to hold for arbitrary values of the arguments of the functions f , g,etc. Under general assumptions on these functions (such as continuity, differentiability,etc.), the imposition that the eq. (0.0.9) should hold for arbitrary values for x, y, . . . , isoften sufficient to almost uniquely fix the functions that solve the functional equation.The notion of initial values is irrelevant in this context.

Exercise # 1: Consider the functional equation:

F(f(x), f(y), f(x + y)) = f(x+ y)− f(x)f(y) = 0 .

Under the assumption that f is differentiable for all real values of its argument, and settingf ′(0) = α 6= 0, show that the unique solution is given by f(x) = eαx .

Exercise # 2: Consider the functional equation:

F (f(x), f(y), g(x), g(y), f(x + y)) = f(x+ y)− f(x)f(y) + g(x)g(y) = 0 .

Under the assumption that f , g are both differentiable for all real values of their argu-ments, and that f is an even function of its argument, determine the general solution of thefunctional equation.

c) Partial difference equations: in this case we have more than one independent variablex, t ∈ R or in the finite difference case n,m ∈ Z. The equations can take the form:

F(y(x− k1δ, t− k2ε), . . . , y(x+ l1δ, t+ l2ε) ;x, t) = 0 (0.0.10)

with k1, k2, l1, l2 fixed integers, in the case of partial analytic difference equations, orin the finite difference case:

F(yn−k1,m−k2 , . . . , yn+l1,m+l2 ;n,m) = 0 . (0.0.11)

The dependent variable y(x, t) and yn,m respectively depend here on two (or more)independent variables with discrete shifts. The second case can be reduced to the firstcase by setting

yn,m = y(x0 + nδ, t0 +mε) ,

but obviously the two types of problems are quite different in nature, since in theanalytic case we want to solve for y(x, t) as a function of a continuous range of values ofits arguments. The theory of the latter type of equations is still very under-developed.

8

δ

ε

−→ n

m↓

One of the amazing facts is that in the study of integrable difference equations, wewill encounter all these type of equations, but moreover these types turn out to be quiteinterconnected. In fact, we will see that in many examnples we can interpret one andthe same system either as a dynamical map or finite difference equation, or as an analyticdifference equation, or even as functional equation, without damaging the main integrabilityaspect of the system. In the latter interpretation we are often dealing with solutions in termsof (nonlinear discrete) special functions.

The subject is also related to a new branch of geometry, called difference geometry, whichis the discrete analogue of the classical differential geometry of curves and surfaces. It turnsout that integrable P∆Es describe discrete quadrilateral surfaces like the one depicted inthe figure below:

The development of this new theory has been one of the more intriguing applications of thesubject.

9

Outline of this Module

In this module we will go through some of the modern developments in the theory of inte-grable difference equations. Basic calculus and algebra are all that are needed as prerequisitesto start with. Later on some knowledge of elliptic functions will be useful (which we willtreat in an elementary way at the advanced level of the module).

We will first start with rehearsing in the first lectures some aspects of the general theoryof difference equations, such as going over some elementary facts from the calculus of finitedifferences, and treating some elements of the theory of linear difference equations as wellas specific basic examples of nonlinear difference equations. After that we will presentsome systematic methods to obtain discrete equations from parameter-families of continuousequations. In particular we will show how certain discrete transformations on the solutionsof differential equations for some well-known special functions lead to difference equationsfor those families of functions. These, in fact, are already manifestations of structures ofintegrable systems.

Next our focus will turn to the theory of partial difference equations, from which we willdevelop the notions of integrability further. To motivate the special partial difference equa-tions under consideration, we will introduce them through the consideration of Backlundtransformations for some of the well-known integrable PDEs. Once we have established theP∆Es, we will study their discrete integrability aspects from a modern perspective. Subse-quently, we will look at a variety of special solutions, which allow us to get a kaleidoscopicview on the various techniques used in integrable systems. The soliton solutions will allowus to introduce a powerful structure in terms of infinite recurrence relations which helpus to find the connections between different integrable equations (this will be done at theadvanced level of the module). Furthermore, these structures can be generalised to obtainP∆Es in higher dimensions related to the famous KP (Kadomtsev-Petviashvili) hierarchy.Other solutions are obtained through the consideration of initial-value problems on the lat-tice, which lead to reductions in terms of integrable dynamical mappings. This brings usto the theory of (finite-dimensional) integrable discrete-time systems. Furthermore, for theadvanced level we will make a deviation into elliptic functions which we need, not onlyto paramatrise solutions of the latter, but also to find yet richer classes of integrable sys-tems. Finally, we will discuss some important structural aspects of the theory, possibly theLagrangian description and/or the structure of symmetries on the lattice and the possiblespecial solutions we can obtain through the corresponsding reduction techniques. The lat-ter would lead us in particular to nonlinear special functions and discrete analogues of thefamous Painleve transcendental equations.

10

General Literature:

1. G.E. Andrews, R. Askey and R. Roy, Special Functions, (Cambridge Univ. Press,1999).

2. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists andEngineers, (McGraw-Hill, 1978).

3. P.R. Garabedian, Partial Differential Equations, (Chelsea, 1964).

4. F.B. Hildebrand, Introduction to Numerical Analysis, (McGraw-Hill, 1956; Dover,1987).

5. E.L. Ince, Ordinary Differential Equations, (Dover, 1956)

6. K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painleve, (Vieweg,1991)

7. L.M. Milne-Thompson, The Calculus of Finite Differences, (AMS Chelsea Publ. 2000),first edition, 1933.

8. N.E. Norlund, Vorlesungen uber Differenzenrechnung, (Chelsea, 1954), first edition ,1923.

9. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, (Cambridge Univ.Press, 1927 (4th ed.))

10. R. Sauer, Differenzengeometrie, (Springer Verlag, 1970).

11. A.S. Bobenko and Yu. Suris,Discrete Differential Geometry; Integrable structure,Grad. Studies in Math. vol 98, AMS Publs. 2008, (a preliminary version can befound on: http:arxiv.org/math/abs/0504358).

12. J. Hietarinta, N. Joshi and F.W. Nijhoff, Discrete Systems and Integrability, (mono-graph in preparation).

Lecture 1

Elementary Theory ofDifference Equations

In this lecture we will give a brief account some of the basic tools to deal with (ordinary)difference equations and systems based on discrete operations. Most of the machinery israther elementary, as we will concentrate here primarily on linear finite-difference equations,although some simple nonlinear equations will be dealt with as well. Some of this materialis needed in subsequent lectures dealing with integrable systems. Some analogies with anddistinctions from the theory of differential equations will be mentioned. Probably the besttext for further reading is the classic monograph by L.M.Milne-Thompson, or the relevantchapters 2 and 5 in the famous book by C.M. Bender & S.A. Orszag. However, thereare many elementary textbooks on the basic theory of finite-difference equations, e.g. thetext by W.G. Kelley & A.C. Peterson. For partial finite-difference equations, material canbe found in the monographs by P.R. Garabedian (chapter 3 and 13) and in the book byF.B. Hildebrand. These latter texts are motivated by the problem of numerical simulationsthrough finite-difference equations.

1.1 Elementary Difference Calculus

1.1.1 Difference versus differential operators

In the Preface we have already introduced the difference operator ∆h, depending on a step-size parameter h, which was defined by its action on a function y(x), of a single independentvariable x through the formula

∆hy(x) =1

h(Th − id) y(x) , (1.1.1)

pointing out that it essentially builds on the shift operator

Thy(x) = y(x+ h) . (1.1.2)

11

12 LECTURE 1. ELEMENTARY THEORY OF DIFFERENCE EQUATIONS

It is tempting to expect that one could build a theory based on the difference operator(1.1.1) in a fairly similar way as for the differential operator d/dx , but there are impor-tant distinctions. Recalling, from elementary calculus, some of the main properties of thedifferential operator, they can be summarised as follows:

linearity: for any two functions y(x), w(x) and constants α, β we have:

d

dx(αy(x) + βw(x)) = α

dy

dx+ β

dw

dx

product rule: For the product of any two fuctions y(x), w(x) we have the Leibniz rule:

d

dx(y(x)w(x)) =

dy

dxw(x) + y(x)

dw

dx

composition rule: for two functions z(y) and y(x) and forming the composed functionZ(x) = z ◦ y(x) = z(y(x)) by substitution we have the chain rule

dZ

dx= z′(y(x))y′(x)

where z′(y) = dz/dy , y′(x) = dy/dx .

For the difference operator, however, the last rule really stops making sense (at least withoutmaking many further assumptions), whilst the product rule takes a subtly differen form,namely by computing

∆h(y(x)w(x)) =1

h[y(x+ h)w(x + h)− y(x)w(x)]

=1

hy(x+ h)[w(x + h)− w(x)] − 1

h[y(x+ h)− y(x)]w(x)

= y(x+ h)∆h(x) + (∆hy(x))w(x) ,

which can be expressed as:

∆h(yw) = (Thy)∆hw + (∆hy)w , (1.1.3)

which looks similar to the differential product rule, but has one crucial extra element: itinvolves necessarily the shift operator Th as well!. Obviously we could write (1.1.4) as wellin the form

∆h(yw) = y∆hw + (∆hy)Thw , (1.1.4)

but in either way the shift operator Th cannot be avoided. Thus, the difference calculuscannot be built with the difference operator ∆h alone, and we need the shift operator Th aswell, which by itself has the crucial product property:

Th(yw) = (Thy)Thw . (1.1.5)

In fact, a ”difference” calculus can be built on the shift operator alone, but we have to realisethat the latter is essentially a nonlocal operator: it connects two values of the argument,

1.1. ELEMENTARY DIFFERENCE CALCULUS 13

namely x and x+h differeing by a fixed stepsize h. Thus, a theory of difference equations, orequations defined by h-shifts is essentially more complicated than the continuous differentialcase, as can also be argued by considering the Taylor expansion: for regular points x of ananalytic function, we have the Taylor series :

Thf(x) = f(x+ h) = f(x) +h

1!f ′(x) +

h2

2!f ′′(x) +

h3

3!f ′′′(x) + · · · (1.1.6)

which shows that the shift operator involves not only the derivative f ′(x) = df/dx, but alsoall higher order derivatives. In fact, the right-hand side of (4.2.1) can be formally expressedas exp(h d/dx) f(x) , showing that the shift-operator Th, is an infinite-order operator.

Let us now consider, in this lecture, the finite-difference case, i.e. we concentrate onfunctions, for the time being only depending on one single discrete variable n which takesvalues in the integers n ∈ Z, and which we can denote by yn. In fact,by setting

yn = y(x0 + nh)

fixing an initial point x0 we can reduce the (analytic) difference situation above by thefinite-difference case in which we have a function of a discrete variable n shifting by units,i.e., for which we can describe the shift by a map: yn 7→ yn+1. This amounts to viewing thefunction y as a map y : N→ R : n 7→ yn, which produces an infinite or semi-infinite sequenceof values: {y−N , y−N+1, . . . , y0, y1, . . . , yn, . . . } . The discrete variable n will be consideredto be the independent variable, whilst the fuction value yn is the dependent variable. Thecalculus we develop in this lecture will concentrate on this particular situation.

1.1.2 The finite-difference operator

For simplicity, we shall for the time being work with the (forward) finite-difference andfinite-shift operators:

(∆y)n = yn+1 − yn := ∆yn , (Ty)n = yn+1 := Tyn , (1.1.7)

and we set T−1yn = yn−1 as being the inverse of the shift operator T . Thus, we have∆ = T − id . Obviously, we could also introduce other types of difference operators, forinstance:

∇yn = yn − yn−1 = (id− T−1)yn, �yn = yn+1 − yn−1 = (T − T−1)yn,

i.e., the backward difference operator and the symmetric difference operator, respectively,but we will make little use of them. Furthermore, the notation varies in the literature andsometimes ∇ is used for the average: ∇yn = yn+1 + yn.

Difference equations can now be written in terms of these operators, and it is indeedoften useful to write linear difference equations in terms of T . It is easy to see that

∆yn = yn+1 − yn = (T − id) yn,

∆2yn = yn+2 − 2yn+1 + yn =(T 2 − 2T + id

)yn,

. . . . . .

∆kyn =

k∑

j=0

(k

j

)(−1)j yn+k−j = (−1)k

k∑

j=0

(k

j

)(−T )k−j yn.


Exercise 1.1.1. Derive the last formula. Show also that

∆∇yn = ∇∆yn = yn+1 − 2yn + yn−1, �2yn = yn+2 − 2yn + yn−2.

As pointed out in the previous subsection, whereas the difference operator ∆ can ob-viously be viewed as a ‘discrete derivative’, analogous to the differential operator d/dx indifferential calculus, there are some crucial differences with the differential case.

We will (almost always) assume that the functions y(x+n) = yn of the discrete variableform a ring of functions (in the usual sense of abstract algebra), and that pointwise algebraicoperations hold (linear combinations, products, etc.). In the same way as derivatives, thedifference operations act naturally on linear combinations of functions:

∆(αxn + βyn) = α∆xn + β∆yn , (1.1.8a)

for two functions x, y : Z→ R, and arbitrary constants α, β.However the product (Leibniz) rule has to be modified from the one for derivatives,

namely:

∆(xnyn) = xn+1yn+1 − xnyn = (Txn)∆yn + (∆xn)yn = (∆xn)Tyn + xn∆yn, (1.1.8b)

noting as we did earlier that one cannot write the product rule in terms of difference operators∆ alone, but one needs to involve the shift operator E as well! This is a crucial distinctionwith the continuous case of derivatives, and the reason why difference equations are usuallymore complicated than differential equations.

Exercise 1.1.2. Compute ∆(yn/zn).

Another crucial distinction is the observation that there is no natural analogue for dif-ference operators of the chain rule for derivatives. In fact, this is because there is no naturalcomposition rule of functions of the discrete variable, unless we specify the target space ofthe maps n 7→ yn in a very special way, namely from integers to integers. However, that iswhat we usually do not want to do, and most commonly we want the functions yn to takevalues in a continuous set, e.g., the reals R or the complex numbers C.

Exercise 1.1.3. Compute the ∆-derivative for the functions log(an), sin(an).

1.1.3 Difference analogues for some familiar functions

The difference operator acts in some ways similar to the derivative (e.g., linearity) but dueto the change in the Leibniz rule many things are also different. Nevertheless we would liketo have analogues to familiar functions, so that ∆ would act on them in a similar way asd/dx.

In particular we would like to have a function on which ∆ would operate as d/dx operatesfor xk, and replace the rule

d

dxxk = kxk−1 ,

by something analogous. If we think of x being replaced by the discrete variable n, such afunction should obey the rule:

∆f(n; k) = k f(n; k − 1) . (1.1.9)


It turns out that the corresponding function can be given in terms of the so-called Pochham-mer symbol (a)k defined by1

(a)k = a (a− 1) (a− 2) . . . (a− k + 1) , for k > 0, (a)0 ≡ 1 , (1.1.10)

in which k is an integer, but where a could still take on any (real) value. This is also calledthe falling factorial2 and it is sometimes denoted by a(k).

Performing the following computation:

∆(n)k = (n+ 1)k − (n)k

= [(n+ 1)n(n− 1) · · · (n− k + 2)]− [n(n− 1) · · · (n− k + 2)(n− k + 1)]

= n(n− 1) · · · (n− k + 2)[(n+ 1)− (n− k + 1)] = k[n(n− 1) · · · (n− k + 2)]

= k (n)k−1,

which shows that f(n; k) = (n)k is a solution of the difference equation (1.1.9) establishingthe analogy with the power function.

Exercise 1.1.4. In the case that k is a negative integer we can introduce the symbol:

(a)k = 1/ ((a+ 1)(a+ 2) · · · (a+ |k|)) k < 0 , (1.1.11)

(also denoted as a(k)). Show that the function f(n; k) = (n)k is a solution of the differenceequation (1.1.9) for the power function also for negative k.

Since ∆ is a linear operator, we can get an analogue to any function defined by a powerseries solution, simply by considering linear combinations of power functions of differentdegree (the degree being teh variable k). For example, the analogue of the exponentialfunction exp(ax) , would be

e(n; a) =

∞∑

k=0

ak(n)kk!

= (1 + a)n. (1.1.12)

Exercise 1.1.5.i) Show that the discrete exponential function e(n; a) obeys the difference equation

∆e(n; a) = a e(n; a) ;

ii) show that the second equality in (1.1.12) holds, and that as a consequence we have therelation

e(n; a)e(m; a) = e(n+m; a) .

1Note that a does not need to be an integer for this definition to make sense. In fact, one can extend thedefinition of (a)k to noninteger values of k by setting

(a)k =Γ(a+ k)

Γ(a)

where Γ denotes the standard Γ function, which we will refrain from defining here. For the time beingit is sufficient to state that Γ(z) is a complex-valued function, generalising the factorial, and obeying thedifference equation: Γ(z + 1) = zΓ(z).

2In the literature appears also the raising factorial, defined by products of increasing factors instead of(1.1.10). To distinguish the two one could introduce the products:

(a)±k

= a (a ± 1) (a± 2) . . . (a ± (k − 1)) ,

but we will not make much use of this notation.


1.1.4 Discrete integration: the antidifference operator

In the same way as the notion of the integral arises as the inverse of the operation of thederivative, i.e., the anti-derivative of the function f(x)):

f(x) =dF (x)

dx⇒ F (x) =

∫ x

f(x′) dx′,

where the indefinite integral denotes a primitive of the function f(x) (fixing a lower limitof integration amounts to specifying an integration constant). The inverse operation of thedifference operator ∆, i.e., its anti-difference, is given by the summation:

fn = ∆Fn ⇒ Fn = ∆−1fn :=

n−1∑

j

fj .

Note that the upper index in the summation is n − 1, since ∆ is defined by the forwarddifferentiation. In both cases the inverse is not unique, but is determined up to a constant:an integration constant w.r.t. the variable x in the derivative case, and a constant w.r.t. thevariable n in the difference case. In the indefinite integration/summation the constant isleft unspecified, while in the definite integration/summation this constant can be fixed byspecifying the lower limit in the integral/sum, e.g.,

F (x) =

∫ x

0

f(x) dx and Fn =

n−1∑

j=0

fj.

This lower limit is purely a matter of choice and the choice depends on the problem athand (e.g., initial values when we deal with differential or difference equations). We can alsodefine the summation so that n is as the lower index:

Gn :=

M∑

j=n

fj , ∆Gn = −fn.

Exercise 1.1.6. Give the antiderivative of the function f(x) = xk, and the antidifferenceof the function f(n; k) = (n)k for both k a positive and negative integer.

The usual rules of linearity apply to the antidifferential:∑

(αxn + βyn) = α∑

xn + β∑

yn. (1.1.13)

There are also rules corresponding to integration by parts:

n−1∑

j=m

fj∆gj = fngn − fmgm −n−1∑

j=m

∆fj Tgj (1.1.14)

The summation symbol can also be used to define a generating function. Let

F (z) :=

∞∑

k=0

fk zk, or F (z) :=

∞∑

k=0

fkzk

k!,


then yn can be recovered as follows:

fn =1

n!

(d

dz

)n

F (z)

∣∣∣∣z=0

=

(d

dz

)n

F (z)

∣∣∣∣z=0

.

This can be useful if the summation can be done to obtain a closed form expression.

1.1.5 ∗ q-Difference operators and Jackson integrals

A case that resides in some sense inbetween is the theory of q-difference equations. Theq-shift and (forward) q-difference operators are defined in a multiplicative way:

Tqf(x) = f(qx) , Dqf(x) =f(qx)− f(x)

(q − 1)x(1.1.15)

with the “base” q being a fixed complex number.A correspondence with the difference case can be made by setting

f(x) = F (ln(x)) , q = eh ⇒ f(qx) = F (ln(x) + h)

but one should be warned that introducing logarithms may affect the analytic nature of thesolutions in a dramatic way.

The power function coincides here with the continuous power function, since

Dqxk = (k)qx

k−1 with (k)q =qk − 1

q − 1(1.1.16)

the (k)q being called q-integers. As a consequence, the so-called q-exponential function canbe defined by

expq(x) =

∞∑

k=0

xk

(k)q!, (k)q! = (k)q(k − 1)q . . . (1)q (1.1.17)

A more common notation is given by the Andrew’s symbols:

(a; q)k =(a; q)∞(qka; q)∞

where (a; q)∞ =∞∏

j=0

(1 − aqj) , |q| < 1 (1.1.18)

which are the q-analogues of the Pochammer symbols. In terms of these symbols the q-exponential can also be defined as eq(x) = 1/(x; q)∞ , since we have the q-binomial identity:

1

(x; q)∞=

∞∑

k=0

xk

(q; q)k= expq

(x

1− q

). (1.1.19)

Exercise 1.1.7. Prove the q-binomial identity (1.1.19) by deriving the difference equationeq(qx) = (1− x)eq(x) , and setting eq(x) =

∑∞

k=0 ckxk derive a recurrence relation for the

coefficients ck, whilst c0 = 1.


The inverse of the q-difference operator Dq is given by the so-called Jackson integral,defined by ∫ x

0

f(x)dqx =: (1− q)x∞∑

k=0

qkf(qkx) (1.1.20)

Exercise 1.1.8. Show that the analogue of the fundamental theorem of integration holds forthe Jackson integral, i.e., that we have

Dq

∫ x

0

f(x)dqx =

∫ x

0

(Dqf)(x)dqx = f(x)

1.2 Linear (finite-)difference equations

We note first, that by defining the anti-difference operators, we have already “solved” somedifference equations, namely equations given in the simple form:

∆Fn = fn , DqF (x) = f(x) with fn, f(x) given.

This would correspond to the simplest linear inhomogeneous case. The next class of differ-ence equations are more general linear difference equations. We know that linear differentialequation are far easier to treat, in general, than nonlinear differential equations, because thelinearity allows the superposition of solutions (at least in the homogeneous case). Similarly,linear difference equation have the same property and, hence, are expected to be simplerthan nonlinear equations. However, both for differential as well as difference equations, onehas to distinguish cases:

• linear homogeneous with constant coefficients;

• linear inhomogeneous with constant coefficients(but possibly with a nonconstant inhomogeneous term);

• linear with nonconstant coefficients.

In the first case the difference equation is autonomous, i.e., the independent variable enters inthe equation only through the dependent variable. For such equations elementary methodsexist to solve them even in explicit form. For the non-constant coefficient case, the situationis much more complicated, and often solutions can only be found through power series, andthe latter are subject to subtle considerations of singular points of the equation where thecoefficients may blow up. In this lecture we will mostly restrict ourselves to considering theconstant coefficient case, but in the next lecture we will come back to the case of non-constantcoefficients.

1.2.1 Linear constant coefficient difference equations

The order of an ordinary differential equation can be defined to be the number of initialdata needed to specify the integration constants in the solution. Thus, a differential equationof the form y′(x) = F (y, x) is of first order, since after separation of variables or othertechniques to find the solution, one integration is needed leading to one free parameter in the

1.2. LINEAR (FINITE-)DIFFERENCE EQUATIONS 19

general solution. hence, we need one initial value to fully specify the solution. An equationof the form y′′ = F (y, y′, x) would be second order, because in general we would needtwo initial data to specify the solution fully, e.g. y(x0), y

′(x0) at a given value x0 of theindependent variable. Similar considerations apply to finite-difference equations as well: theorder of such an equation would by definition equate the number of initial values needed tospecify fully the solution.

Example: A 2-point equation of the form ∆yn+1 = F (yn) would be generically of first order,

since we could start from a given value y0 and iterate the solution as a map, thereby fixing the

solution entirely. However, it is not true that for all functions F defining the equation we would have

a first order difference equation, because if we take F (y) := c−y, then the difference equation would

reduce to: yn+1 = c , c being a constant, and the solution of this equation is simply yn ≡ c, ∀n.The latter would effectively be a 1-point equation, and hence an equation of zeroth order (no initial

values needed to specify the solution, taking into account that c is a constant of the equation, and

hence given a priori). Similar arguments apply to a difference equation of the form yn+1 = G(yn) ,

which is obviusly connected to the previous form by setting G(y) = F (y) + y.

Example: A 3-point equation of the form yn+1 + yn−1 = F (yn) would be generically of secondorder, since we would for generic F need two initial date, say y−1, y0 to iterate the equation as amap:

(yn−1, yn) 7→ (yn, yn+1) = (yn,−yn−1 + F (yn))

However, it is not alway right to consider the digits in the shifts of a difference equation as an

indication of the order. Thus, the equation ∆2yn = F (yn+1, yn) does not always need to be of

second order, even though we have a second degree difference operator involved. In fact, if we take

F (y, y) = f(y)− 2y, then the equation reduces to a 2-point form, and the solution would be of first

order, albeit on a sublattice of integers of either even or odd labeled points.

The general case of first order, constant-coefficient, linear finite-difference equations is notvery exciting. Such equations can be written in the form ayn+1+byn = 0 , (a, b constant) inthe homogeneous case, leading to the solution yn = (−b/a)ny0 , and in the inhomogeneouscase (with given function fn) we can follow the procedure of the example below.

Example: Solve the inhomogeneous first order difference equation

ayn+1 + byn = fn , a, b constants , fn a given function of n.

The solution of the homogeneous equation (replacing fn by 0) is given above. By a principle whichis called variation of constants, we set yn = ((−b/a)nwn (i.e. replacing the constant y0 by afunction wn of n yet to be determined). Inserting this into the equation yields:

(− b

a

)n [a

(− b

a

)wn+1 + bwn

]= fn ⇒ wn+1 − wn = −1

b

(−a

b

)n

fn ,

and the latter equation can be “integrated” by a summation, yielding:

wn = w0 −1

b

n−1∑

j=0

(−a

b

)j

fj , w0 (constant) intial value .


Thus, returning to the variable yn, the general solution is given by:

yn =

(− b

a

)n

w0 −1

b

n−1∑

j=0

(− b

a

)n−j

fj ,

in which w0 plays the role of an “integration constant”.

Example: We will now consider the general case of second order constant-coefficient, linearfinite-difference equations, the general form of the equation being:

ayn+1 + byn + cyn−1 = fn , a, b, c constants , fn a given function of n. . (1.2.21)

If fn ≡ 0 then the equation is called homogeneous, otherwise it is called inhomogeneous. Theprocedure for finding the solution follows closely the standard method for solving constant coefficientdifferential equations, comprising the following steps:

1. Find the general solution of the corresponding homogeneous equation:

ayn+1 + byn + cyn−1 = 0 .

This requires solving the corresponding characteristic equation associated with this differenceequation, which is obtained by exploring a simple trial solution of the form yn ∼ λn , whereλ is to be determined. Inserting this into the euqation we get the characteristic equation:

aλn+1 + bλn + cλn−1 = 0 ⇒ aλ2 + bλ+ c = 0 .

Then we distinguish the following possibilities:

-distinct roots, λ1 6= λ2: the general solution in this case is yhomn = c1λ

n1 + c2λ

n2 ;

-coinciding roots, λ1 = λ2: the general solution is given by yhomn = (c1 + c2n)λ

n1 ,

where c1, c2 are arbitrary parameters (integration constants) of the solution3.

2. Find a specific solution of the inhomogeneous equation, e.g. by educated guess, or, moresystematically by the method of variation of constants (see problem on the Examples # 1sheet: this amounts to allowing c1, c2 in the general homogeneous solution to depend on nand by plugging this into the inhomogeneous equation to find the correpsonding c1(n), c2(n)).We note that in general, a particular solution of the inhomogeneous equation (1.2.21) is givenby:

ypartn =

1

a

n−1∑

j=0

λn−j2 − λn−j

1

λ2 − λ1fj , (1.2.22)

when λ1 6= λ2.

3Note that these solutions are the consequence of the superposition principle: in the case of distinctroots λn

1 and λn2 are two independent solutions of the homogeneous equation, and the general solution is an

arbitrary linear combination of these two. In the case of coinciding roots λ1 = λ2 the independent solutionsare λn

1 and nλn1 and again a linear combination of these two solutions leads to the general solution. The

independent solutions in the latter case can be seen as arising from a limiting procedure of the former case:

replacing the independent solutions λn1 , λ

n2 w.l.o.g. by λn

1 andλn2 −λn

1λ2−λ1

, (note that the latter is itself a linear

combination of the two orginal independent solutions), and taking the limit that the two roots coincide,i.e. λ2 = λ1 + ǫ, with ǫ → 0, the second solution goes in this limit over into nλ

n−11 , which is the other

independent solution (up to a constant factor) in the coinciding roots case.


Exercise 1.2.1. Verify that the formula (1.2.22) yields a particular solution of (1.2.21).Give a similar formula for the case of coinciding roots λ1 = λ2 by setting λ2 = λ1 + ǫ, andtaking the limit ǫ → 0.

3. The general solution of the inhomogeneous equation is now given by the combination yn =yhomn + ypart

n of the general solution of the homogeneous equation (as given in step # 1) andthe particular solution of the inhomogeneous equation (as given in step # 2). The solutiondepends on integration constants c1, c2, which can subsequently be determined (but onlyonce we have the full solution of the inhomogeneous problem) by imposing initial conditions,e.g. by giving y0 and y1.

Example: The Fibonacci numbers are given by the recursion relation

Fn+1 = Fn + Fn−1, or (T 2 − T − 1)Fn = 0,

with initial values F0 = F1 = 1. The characteristic equation is given by

λ2 − λ− 1 = 0 ⇒ λ =1

2(1±

√5)

and hence the general solution

Fn = a[1

2(1 +

√5)]n + b[

1

2(1−

√5)]n

F0=F1=1=⇒ a =

1

2√5(1 +

√5) , b = − 1

2√5(1−

√5).

Alternatively, we can use a generating function to find the Fibonacci numbers: Introduce F (z) =∑∞

n=0 Fnzn , with z an indeterminate. Plug this into the equation to get:

∞∑

n=0

(Fn+2 − Fn+1 − Fn) zn = z−2[F (z)− F1z − F0]− z−1[F (z)− F0]− F (z) = 0

Use the initial conditions F0 = F1 = 1 to solve for F (z), yielding:

F (z) =F0(1− z) + F1z

1− z − z2=

1

1− z − z2

and expanding the r.h.s. w.r.t. powers of z yields the Fibonacci numbers as coefficients.

1.2.2 Linear first order ∆Es with nonconstant coefficients

Let us now consider the general case of non-constant coefficient first-order linear differenceequation, which can be treated to by a technique analogous to the method of integratingfactors in differential equations. Thus, let us consider a difference equation of the type:

yn+1 − gnyn = hn, (1.2.23)

where gn(6= 0), hn are some given functions. As before, the complete solution of (1.2.23)is a sum of the general solution of the associated homogeneous problem and a particularsolution of the full inhomogeneous equation.

It is easy to see, that the homogeneous version of (1.2.23), namely

wn+1 − gn wn = 0 , (1.2.24)


can be solved by

wn = w0

n−1∏

j=0

gj , (1.2.25)

and, thus, by setting yn = wnzn , where a new (unknown) function zn remains to bedetermined, we obtain by substitution the difference equation

zn+1 − zn = hn/wn+1 . (1.2.26)

Since the left-hand side is now of the form ∆zn, the equation can be simply integrated withas solution

zn :=

n−1∑

k=k0

hk/wk+1, (1.2.27)

where k0 is an arbitrary lower limit of the sum. Thus the general solution of (1.2.23) is

yn :=

(C +

n−1∑

k=k0

hk∏kj=j0

gj

)n−1∏

j=j0

gj . (1.2.28)

Here C is the “summation constant”, associated with the homogeneous solution, its valuebe determined by an initial condition. This formula (1.2.28) is really only a formal solution,and in practice we would like to express the sums and products in closed form, but thereare no general methods for doing that. Sometimes it is useful to note that

∏gj = e

∑log gj .

Example: In order to solve

yn+1 − nyn = n,

we use (1.2.28) with gn = n, hn = n. For convenience we take j0 = k0 = 1. Then we get

yn = (n− 1)!(C +

∑n−1k=1

1(k−1)!

).

1.2.3 Higher order linear difference equations and linear (in)dependence

Most of the above considerations can be extended to the higher order case as well. Thegeneral linear equation is of the form

a(N)n yn+N + a(N−1)

n yn+N−1 + · · · a(1)n yn+1 + a(0)n yn = fn (1.2.29)

where a(j)n , fn are some given functions of n, and where we assume that the product aNn a

0n is

nonvanishing. In the case of constant coefficients (i.e., when all a(j)n = a(j) do not depend on

the independent variable n) the homogeneous problem reduces to solving the correspondingcharacteristic equation:

a(N)λN + a(N−1)λN−1 + · · · a(1)λ+ a(0) = 0 ,


leading to roots λ1, . . . , λN (some of which may coincide). If all roots are mutually distinct,the solution would take the form

yn =

N∑

j=1

cjλnj , cj constant coefficients

but if some roots coincide we will have to supply different independent solution. In fact, ifa root λj has multiplicity µj (i.e. µj is the power of the factor (λ− λj) in the factorisationof the polynomial given by the characteristic equation), then the corresponding linearlyindependent solutions are λnj , nλ

nj , . . . , n

µj−1λnj . The general solution of (1.2.29) in theinhomogeneous case is given by a linear combination of N linearly independent generalsolutions of the homogeneous part plus a particular solution of the inhomogeneous equation.

In the case of non-constant coefficients (i.e., when the coefficients (1.2.29) a(j)n are non-

trivial functions of n) the situation is markedly more complicated, and to find basic solutionsone needs to resort to developing power series in the same way as is done in the Fuchsiantheory of differential equations. Such power series solutions often depend crucially on theoccurrence of singularities of the coefficients. We will refrain from treating this theory atthis juncture. The only issue we want to mention at this point is the nontrivial matter oflinear independency of solutions of the difference equation. In the continuous case of lineardifferential equations the linear independence of solutions is expressed through the non-vanishing of a particular determinant associated with the differential equation, the so-calledWronski determinant or Wronskian (see Example sheet # 1). In the case of linear differ-ence equations the role of the Wronskian is replaced by the so-called Casorati determinantor Casoratian. This determinant is constructed as follows. Suppose that we have N discrete

functions y(j)n , j = 1, . . . , N , then the Casoratian of these functions is given by

C(y(1)n , y(2)n , . . . , y(N)n ) =

∣∣∣∣∣∣∣∣∣∣

y(1)n y

(2)n . . . y

(N)n

y(1)n+1 y

(2)n+1 . . . y

(N)n+1

......

. . ....

y(1)n+N−1 y

(2)n+N−1 . . . y

(N)n+N−1

∣∣∣∣∣∣∣∣∣∣

, (1.2.30)

and these functions are independent if and only if their Casoratian is nonzero.

Exercise 1.2.2. In the homogeneous case of (1.2.29), i.e., setting fn ≡ 0, show that then-dependence of the Casoration is given by:

C(y(1)n+1, y

(2)n+1, . . . , y

N)n+1) = (−1)N a

(0)n

a(N)n

C(y(1)n , y(2)n , . . . , yN)n )

which, provided a(0)n /a

(N)n 6= 0 , guarantees that the indpendence of the solutions is preserved

as n progresses.


The question of solutions of linear nonconstant coefficient dif-ference equations becomes even more pressing in the theory oflinear analytic difference equations, i.e. the case that the inde-pendent variable can take on all real or complex values (see thediscussion in the Preface). The latter linear theory was devel-oped in the beginning of the 20th century by the school of GeorgeD. Birkhoff and collaborators (W.J. Trjitzinsky, R. Carmichael,J. LeCaine). This group of mathematicians developed system-atic methods for the Fuchsian theory of linear difference equa-tions (some of which is represented in the monograph by N.E.Norlund, cited in the Preface). Unfortunately, the work in thisdirection seemed to have ceased with the onset of the secondworld war. It was only in recent years that aspects of the re- Figure 1.1: G.D. Birkhoff

sults of the Birkhoff school were reconsidered in modern mathematics, and various groupsare currently in the process of finalising the work of this school (e.g. the group of Prof.Jean-Pierre Ramis and his students at Toulouse University).

∗ Variation of constants – matrix formulation

We first note that the Nth order linear difference equation (1.2.29) can be cast into a systemof N linear first order equations by introducing new variables

y[j]n =: yn+j−1 ⇒ y[j]n+1 = y[j+1]

n , j = 1, . . . , N − 1

and furthermore

y[N ]n =: yn+N−1 ⇒ y

[N ]n+1 = yn+N =

1

a(N)n

(fn − a(N−1)

n y[N ]n − · · · − a(1)n y[2]n − a(0)n y[1]n

).

Introducing the n-component vector yn =(y[1]n , y

[2]n , . . . , y

[N ]n

)Twe can now write the

system of equations above, in vector form as follows:

yn+1 = Anyn + fn , (1.2.31)

with An a matrix of coefficient, and fn a vector given by:

fn =

00...

fn/a(N)n

, An =

0 1 0 . . . . . . 00 0 1 0 . . . 0...

. . .. . .

. . ....

0 1

− a(0)n

a(N)n

− a(1)n

a(N)n

. . . . . . −a(N−2)n

a(N)n

a(N−1)n

a(N)n

.

The general solution of the matrix difference equation (1.2.31) is very easily obtained, oncewe assume that we know a fundamental matrix solution Y n of the homogeneous matrixequation, i.e., a N ×N matrix solution of the equation

Y n+1 = AnY n , such that det(Y n) 6= 0.


Note that any matrix Y n obtained from Y n by matrix multiplication from the right by aninvertible constant matrix C, i.e.,

Y n → Y n = Y nC

is again a fundamental matrix solution of the same homogeneous matrix difference equation.Whether or not the sum can be computed in closed form depends on the particular natureof the coefficients in the equation and whether or not we can find a reasonable expressionfor the fundamental matrix solution Y n.

The variation of constants approach now amounts to finding a solution of the inhomo-geneous vector equation (1.2.31) of the form yn = Y n cn where the N -component vectorcn remains to be determined as a function of n. Plugging this form of yn into the inhomo-geneous equation we obtain:

Y n+1cn+1 = AnY ncn+1 = AnY ncn + fn

⇒ cn+1 − cn = Y −1n+1fn

⇒ cn = c0 +

n−1∑

j=0

Y −1j+1f j

⇒ yn = Y ncn = Y nc0 +

n−1∑

j=0

Y nY−1j A−1

j f j .

Thus, once Y n is known, the last formula provides the solution of the vector differenceequation, and this is also the general solution, because the first term contains an arbitraryconstant N -component vector c0 which corresponds to the N integration constants of theequation.

In the case of the Nth order difference equation (1.2.29) the vector yn can be written as

yn = (yn, yn+1, . . . , yn+N−1)T

and if we know N independent solutions of that equation y(1)n , . . . , y

(N)n the matrix Y n is

given by

Y n =

y(1)n y

(2)n · · · y

(N)n

y(1)n+1 y

(2)n+1 · · · y

(N)n+1

......

...

y(1)n+N−1 y

(2)n+N−1 · · · y

(N)n+N−1

from which it is clear that the determinant det(Y n) can be identified with the Casoratian ofthe N independent solutions. The explicit case of N = 2 of this construction will be studiedin the Examples # 1 sheet.

1.2.4 Further techniques

1. Factorisation method: It is sometime useful to write the equation in terms of the Eshift operator

(aNn TN + aN−1

n TN−1 + · · · a1nT + a0n)yn = hn, (1.2.32)

and try to factorize the operator.


Example: The difference equation

yn+1 + yn−1 =2n2

n2 − 1yn

can also be written as

(T 2 − 2n2

n2−1T + 1)yn−1 = 0.

The operator factorizes as

(T − n

n−1)(T − n−1

n)yn−1 = 0.

First let

wn−1 := (T − n−1n

)yn−1,

and solve

(T − n

n−1)wn−1 = 0, =⇒ wn = 3Cn.

Next

(T − n−1n

)yn−1 = 3C(n− 1)

is solved with

yn = C(n2 − 1) +D/n.

2. Reduction from homogeneous to inhomogeneous equation: If a particular solu-tion of the inhomogeneous equation can be found that it can be used to reduce the probleminto an homogeneous one.

Another method of reducing the problem to a homogeneous one is to operate on theequation with an operator that annihilates the inhomogeneous part. Then one solves theresulting (higher order) homogeneous equation. The solution is then substituted into theinhomogeneous equation and the extra coefficients determined.

3. Reduction of order: With a known solution of the homogeneous equation the orderof the equation can be reduced.

Example: Consider the second order equation

a2yn+2 + a1yn+1 + a0yn = b, ∀n,

where ai, b may depend on n. Suppose y[1]n is a solutions of the homogeneous part. Then substituting

yn = y[1]n zn yields

a2y[1]n+2(zn+2 − zn+1) + (a2y

[1]n+2 + a1y

[1]n+1)(zn+1 − zn) = b

which is first order in wn := zn+1 − zn.

1.3. A SIMPLENONLINEAR DIFFERENCE EQUATION – DISCRETE RICCATI EQUATION27

4. z-transforms: In treatment of the Fibonacci difference equation we have introducedthe generating function of the Fibonacci numbers. This is a particular example of a z-transform for the difference equation (reminiscent of the Laplace transform for differentialequations): by introducing suitable series expansions in terms of an auxiliary variable z,such as

Y (z) =∞∑

j=0

yjzj ,

the linear difference equation can be often converted into an algebraic equation for Y (z) (inthe case of constant coefficients), or into an ODE (for coefficients depending polynomialyon the independent discrete variable n). Examples are given on the Example sheet # 1.

1.3 A simple nonlinear difference equation – discreteRiccati equation

Sometimes a nonlinear difference equation can be linearized, i.e. rendered into a linearequation by a suitable change of variables or by applying other tricks. A case where thishappens is the important example of the discrete Riccati equation, i.e., a difference analogueof the famous Riccati differential equation, named after the Count Jacopo Francesco Riccati(1676-1754). Since we will often encounter both the discrete as well as the continuous Riccatiequation in the context of integrable systems, we will present here a few properties of bothequations.

1.3.1 Continuous Riccati equation

The usual Riccati differential equation is a first order nonlinear differential equation of theform:

dy

dx= a(x)y2 + b(x)y + c(x) , (1.3.33)

i.e., it possesses a quadratic nonlinearity. The coefficients a(x), b(x) and c(x) can be arbitraryfunctions of x. In the case they are constants, the equation (1.3.33) can be solved byseparation of variables.

Exercise 1.3.1. In the case that the coefficients a,b,c in (1.3.33) are constants, and denotingby y1, y2 the roots of the quadratic ay2+ by+ c , give an expression for the general solutionin terms of y1, y2. What happens if the roots of this quadratic coincide?

In the general case, when the coefficients are not necessarily constant, we note the fol-lowing properties:

1. the Riccati equation is form invariant under linear fractional transformations (Mobiustransformations):

y(x) 7→ y(x) =αy(x) + β

γy(x) + δ⇔ y(x) =

δy(x)− β−γy(x) + α

, (1.3.34)


where α, β, γ, δ are functions of x such that αδ − βγ 6= 0. In fact, it is easy to checkthat implementing the transformation (1.3.34) turns (1.3.33) into an equation for y ofthe form

dy

dx= a(x)y2 + b(x)y + c(x) .

with new coefficients a(x), b(x), c(x).

2. The Riccati equation can be linearized by setting: y(x) = f(x)/g(x), substitution ofwhich leads to

gf ′ − fg′ = af2 + bfg + cg2 .

and the latter relation can be split into two linear equations, leading to the system:

{f ′ = λf + cg ,g′ = −af + (λ− b)g ,

where λ is arbitrary. Choosing λ in appropriate way and eliminating g, this systemleads to a second-order linear ODE for f .

3. Any four solutions yi(x) , (i = 1, 2, 3, 4) of the same Riccati equation are relatedthrough the relation

CR[y1, y2, y3, y4] =:(y1 − y2)(y3 − y4)(y1 − y3)(y2 − y4)

= constant , (1.3.35)

where CR stands for cross-ratio. In fact, for any two solutions yi, yj of the Riccatiequation, it is easy to derive:

d

dxln(yi − yj) = a(yi + yj) + b ,

and, hence, the combination

d

dxln((y1 − y2)(y3 − y4))

is invariant under permutations of the indices.

4. Given two solution y1(x), y2(x) of the same Riccati equation, a new solution can befound in the “interpolation form”:

y(x) =y1 + ρy21 + ρ

provided ρ(x) solvesdρ

dx= a(y1 − y2)ρ . (1.3.36)

Hence, given any two solutions, we can find an interpolating solution of the Riccatiequation through a solution of the linear equation for ρ.

Exercise 1.3.2. Prove the statement in part 4 above, deriving the linear differential equationfor ρ in (1.3.36), given two solutions y1, y2 of the Riccati equation.

1.3. A SIMPLENONLINEAR DIFFERENCE EQUATION – DISCRETE RICCATI EQUATION29

1.3.2 Discrete Riccati equation

The Riccati equation (1.3.33) has a natural analogue, the discrete Riccati equation given by

ynyn+1 + anyn+1 + bnyn + cn = 0 , (1.3.37)

where the coefficients an, bn, cn are given functions of n. In fact, it is more natural tointroduce the form:

R(yn, yn+1) = Cnynyn+1 +Dnyn+1 −Anyn −Bn = (−1, yn+1)

(An Bn

Cn Dn

) (yn1

).

(1.3.38)and denote by R(yn, yn+1) = 0 the discrete Riccati equation. It follows immediately fromthis form that the discrete Riccati equation can be linearised as:

(yn+1

1

)= λ

(An Bn

Cn Dn

) (yn1

)

with λ some proportionality constant. In fact, λ = (Cnyn +Dn)−1 Furthermore, solving

for yn+1, we obtain the fractional linear form:

yn+1 =Anyn +Bn

Cnyn +Dn,

which reminds us of the Mobius transformation. In other words, the Mobius transform,interpreted as a map yn 7→ yn+1 is equivalent to the iteration of solutions of the Riccatiequation.

We recover the differential Riccati equation (1.3.33) from the discrete one in the form(1.3.38) performing the limit ε→ 0, setting

yn+1 ∼ y(x) + εdy

dx,

Cn

Dn∼ −εa(x) ,

An −Dn

Dn∼ εb(x) ,

Bn

Dn∼ εc(x)

Exercise 1.3.3. Show that from the limit described above one indeed recovers the continuousRiccati equation in the form (1.3.33).

Let us now consider again the discrete Riccati equation in the form (1.3.37). We willnow establish analogous properties of this equation to ones of the continuous equation.

1. The equation is form-invariant under fractional linear (Mobius) transformation. Infact, under the transformation

yn 7→ yn =αyn + β

γyn + δ, αδ − βγ 6= 0 ,

the equation (1.3.37) goes over into an equation fr yn of the same form with coefficients

an, bn, cn.

2. Given two independent solutions y(1)n , y

(2)n the combination

yn =y(1)n + ρny

(2)n

1 + ρn


is again a solution of the discrete Riccati equation provided that ρn solves the lineardifference equation

(y(2)n y(1)n+1 + any

(1)n+1 + bny

(2)n + cn)ρn + (y(1)n y

(2)n+1 + any

(2)n+1 + bny

(1)n + cn)ρn+1 = 0 .

By using again the discrete Riccati equation for y(1)n , y

(2)n , we can deduce from this

that this linear equation can be written in either of the two equivalent forms:

ρn+1 =y(2)n + an

y(1)n + an

ρn =y(1)n+1 + bn

y(2)n+1 + bn

ρn .

3. For any two different solutions y(i)n and y

(j)n we have the following relation

y(i)n+1 − y

(j)n+1

y(i)n − y(j)n

=cn − anbn

(an + y(i)n )(an + y

(j)n )

,

and hence we conclude that

(y(1)n+1 − y

(2)n+1)(y

(3)n+1 − y

(4))n+1)

(y(1)n − y(2)n )(y

(3)n − y(4))n )

=(cn − anbn)2

(an + y(1)n )(an + y

(2)n )(an + y

(3)n )(an + y

(4)n )

,

for any four solutions y(1)n , y

(2)n , y

(3)n , y

(4)n . Consequently, we infer immediately that for

any four solutions y(1)n , y

(2)n , y

(3)n , y

(4)n , of the discrete Riccati equation we have that

the cross-ratio

CR[y(1)n , y(2)n , y(3)n , y(4)n ] =(y

(1)n − y(2)n )(y

(3)n − y(4)n )

(y(1)n − y(3)n )(y

(2)n − y(4)n )

= constant .

4. By setting yn = fn/gn in (1.3.37) we can show that the equation can be linearisedleading to the linear matricial equation:

(fn+1

gn+1

)= κ

(bn cn−1 −an

)(fngn

), (1.3.39)

in which κ can be chosen arbitrarily. It is easy to derive from this matricial equationa second order linear difference equation for fn.

Exercise 1.3.4. Derive the matrix equation (1.3.39) from the discrete Riccati equation(1.3.37), and by eliminating gn derive a second order difference equation for fn.

Example: An alternative, more direct way to linearize the Riccati equation (1.3.37) is by rewrit-ing the equation as

(yn + an)(yn+1 + bn) + cn − anbn = 0 , (1.3.40)

which suggests a substitution of the form

yn + an =wn

wn−1⇒ wn+1 + (bn − an+1)wn + (cn − anbn)wn−1 = 0 ,

which brings us directly to a linear second order difference equation for wn.

In some isolated cases a judicious grouping of terms can help to find a linearizing trans-formation.

1.4. PARTIAL DIFFERENCE EQUATIONS 31

Example: Consider the difference equation

yn+1yn−1 − y2n = ynyn−1.

Dividing by ynyn−1 yieldsyn+1

yn− yn

yn−1= 1,

which is linear for wn = ynyn−1

.

Exercise 1.3.5. Linearize the logistic equation (Verhulst equation) dy/dx = ay(1− y) bysetting y = 1/(w + b). Do the same for a discrete analogue of the logistic equation, namelyyn+1 = ayn(1 − yn+1) . Show also that the linearization trick breaks down for anotherdiscretization of the logistic equation, namely yn+1 = ayn(1 − yn) .

1.4 Partial Difference Equations

Partial difference equations ( P∆E) have arisen in the context of numerical methods forPDEs, where they correspond to finite-difference schemes on lattices, cf. the monographsby P.R. Garabedian and F.B. Hildebrand. In this section we will mention briefly someaspects, but we will not go into these methods in any depth because the type of P∆Es weare interested in this course arise from quite different considerations and require their ownspecific methodology.

1.4.1 Discretization and boundary values

An the first example let us consider the heat equation

∂y

∂t=∂2y

∂x2.

If we discretize the time derivative using the forward difference ∆ and the second orderspace derivative with ∆∇ we obtain

1ε (y(x, t+ ε)− y(x, t)) = 1

δ2 (y(x+ δ, t)− 2y(x, t) + y(x− δ, t)),

or with discrete indices identifying yn,m = y(x0 + nδ, t0 +mε), relative to some arbitraryorigin (x0, t0), we can then write this equation more simply as:

yn,m+1 = α(yn+1,m + yn−1,m) + βyn,m, (1.4.41)

where α = ε/δ2, β = 1− 2ε/δ2. The points that are involved in this equation are indicatedin Figure 1.2

An important problem in solving P∆E’s is the boundary or initial value question. Thisoften requires a careful consideration of graph of the lattice points involved in the equation.In the present case we can see from Figure 1.2 that given data on the line t = 0, i.e., m = 0we can propagate the solution forward in time. If instead we want to propagate into thepositive x-direction, we need initial data on two vertical lines, say n = n0, n0 +1. We could


t m

x n

a)

t m

x n

b)

Figure 1.2: Points involved a) in equation (1.4.41) and b) in (1.4.42).

also use two lines on an angle, such as m − n = s0, s0 + 1. These initial values correspondto the equation being second order in the x derivatives.

For practical calculations it is often useful to introduce two shift operators,

Tyn,m = yn+1,m, Syn,m = yn,m+1.

For example the discrete heat-equation (1.4.41) can then be written as

Syn,m = (α+ β + αT−1)yn,m.

If we have now the initial value given as yn,0 = fn then we find the solution in the form

yn,m = (αT + β + αT−1)mfn.

Next consider the Laplace equation

∂2y

∂t2+∂2y

∂x2= 0.

This could be discretize as the second derivative in the heat equation, but in both x and t.If we keep the same central point in both discretizations we obtain

yn,m+1 + yn,m−1 + σ(yn+1,m + yn−1,m) + ρyn,m = 0, (1.4.42)

where σ = ε2/δ2, ρ = −2(1 + ε2/δ2). The points that are involved in this equation areindicated in Figure 1.2. From the Figure it is now clear that we need can proceed if we aregiven the initial values on two adjacent vertical lines or horizontal lines.

1.4.2 Separation of variables

For linear constant coefficient PDE’s the ansatz y(x, t) = f(ax+bt) often leads to a solutionif the parameters a, b are chosen suitably. If the terms in the equation have the same totalorder we get a characteristic equation for the parameters, while function f can be arbitrary,in other cases we can reduce the equation to an ODE.

In discrete case the above works much less frequently. It is useful if the points involvedfall on a line, we can then change to new variables and reduce the problem to an O∆E.

1.4. PARTIAL DIFFERENCE EQUATIONS 33

Example: Consider the equation

yn,m = ryn+1,m−2 + syn+2,m−4.

If we now defined new variable as follows:

k = n, l = 2n+m

the equation becomes

yk,l = ryk+1,l + syk+2,l

which can be considered as an O∆E in k. Let now fi(k) be the two solutions of this equation (asdiscussed in Section 1.2.1), then the solution of the original equation is

yn,m = h1(2n+m)f1(n) + h2(2n+m)f2(n),

where the hi are arbitrary functions.

Other classes of mostly nonlinear P∆Es will be considered in subsequent lectures mo-tivated by the context of integrable systems. Their stencil on the lattice can take differentshapes, e.g. of a four-point hook form:

H(yn,m, yn+1,m, yn+1,m−1, yn+2,m−1) = 0 (1.4.43)

a four-point quadrilateral form:

Q(yn,m, yn+1,m, yn,m+1, yn+1,m+1) = 0 (1.4.44)

or a five-point star form:

S(yn,m, yn+1,m, yn−1,m, yn,m+1, yn,m−1) = 0 (1.4.45)

or even more exotic shapes.The points that are involved in the above two-dimensional maps are given in Figure 1.3.

a)

m

nε

δ

b) c)

Figure 1.3: A difference equation may involve a different configuration of points in thelattice. The cases a), b) and c) correspond to (1.4.43), (1.4.44) and (1.4.45), respectively


ց

ր?

Figure 1.4: Acceptable and problematic initial values for (1.4.44)

From Figure 1.4 we can also infer that the definition of an initial value problem can bedelicate. For example in case b), which is discussed in detail later, we can propagate fromany staircase-like initial values, but any “overhang” could lead to a contradiction in theinitial-value problem (due to overdeterminedness).

The problem of establishing continuum limits is also more complicated for P∆E’s thanfor O∆E’s, and in fact not unique: there are more than one way of obtaining contiuousequations from discrete ones by limits. We return to this problem later, at this point itsuffices to note that sometimes we may have to take the continuum limit along a diagonalof the lattice in order to obtain nontrivial equations.

Literature:

1. L.M. Milne-Thompson, The Calculus of Finite Differences, (AMS Chelsea Publ. 2000),first edition, 1933.

2. C.M. Bender and S.A. Orszag: “Advanced Mathematical Methods for Scientists andEngineers”, (McGraw-Hill, 1978)

3. W.G. Kelley and A.C. Peterson: “Difference Equations: An Introduction with Appli-cations”, Second Ed. (Academic Press, 2001).

4. P.R. Garabedian, Partial Differential Equations, (Chelsea, 1964).

5. F.B. Hildebrand, Finite-Difference equations and Simulations, (Prentice-Hall, 1968).

6. G. Andrews, R. Askey and R. Roy, Special Functions, (Cambridge Univ. Press, 1999).

Lecture 2

From Continuous to DiscreteEquations via Transformations

In this part we will discuss how difference equations arise from transformations applied todifferential equations. Thus we find that some functions can be defined both by a differentialequation and by a difference equation. Of course the independent variables of these two typesof equations are not the same, but rather we find interesting duality between parametersand independent variables.

Often functions defined by a differential equation posses special transformations. Oneparticularly important class of transformations are those that act on the differential equationby changing the values of its parameters. When such transformations are iterated, we obtaina sequence of differential equations (and often a sequence of solutions) that are strungtogether by the changing sequence of parameter values.

As particularly simple example consider the differential equation

xw′ = αw. (2.0.1)

Here x is the independent (continuous) variable, the prime stands for x-derivative (w′ = dwdx )

and α is a parameter of the equation. It is easy to find the solution of this equation, it is

wα(x) = k xα, (2.0.2)

where k is required integration constant. From the form of the solution (2.0.2) we can see,that the simple transformation of multiplying the function wα by x changes α, that is

wα+1 = xwα. (2.0.3)

Equation (2.0.3) can now be interpreted as a discrete or difference equation, where α isthe (discrete) independent variable and x is a parameter. Thus we have found, that theequations (2.0.1) and (2.0.3) both describe the same function wα(x) of (2.0.2) but fromtwo different points of view. The key observation here is that of the compatibility betweenthe differential equation (2.0.1) and the difference equation (2.0.3) which can be explicitly

35

36LECTURE 2. FROMCONTINUOUS TODISCRETE EQUATIONS VIA TRANSFORMATIONS

observed from the following litte computation:

w′α+1 = wα + xw′

α ⇒ xw′α+1 = xwα + x2w′

α

⇒ (α+ 1)wα+1 = xwα + x(αwα) ⇔ wα+1 = xwα ,

which could also be stated as saying that the operation of taking the derivative d/dx andthat of the shift in the parameter α, Tα, commute on solutions, i.e., (Tαw)

′ = Tα(w′) . This

is a theme we will see coming back many times.We will first discuss the case of such procedures for ordinary differential equations

(ODEs), using the examples of equations for certain classical special functions, such asthe Bessel, Legendre, and hypergeometric functions. These can be shown not only linearODEs, but also some ordinary difference equations (O∆Es) in terms of certain parameters.There are also nonlinear special functions, such as the elliptic functions and the Painlevetranscendents, that satisfy nonlinear differential equations, for which similar ideas apply,but we will not treat them in this Lecture. Next, we will move from ODEs to partial differ-ential equations (PDEs), and treat briefly some basic theory of soliton equations, and showhow similar ideas there lead to the construction of integrable partial difference equations(P∆Es).

2.1 Special Functions and linear equations

Special functions [see, e.g., Whittaker-Watson, 1927] such as the Bessel, Parabolic Cylin-der, and hypergeometric functions satisfy ODEs with an independent variable x and atleast one parameter α. They also satisfy recurrence relations in which the parameter α isshifted. We will now study some of these differential equations and derive the correspondingtransformations and related difference equations.

2.1.1 Weber Functions and Hermite Polynomials

Consider the differential equation

w′′ +(α+ 1

2 − 14x

2)w = 0, (2.1.4)

where the primes denote differentiation in x. This equation arises in many physical applica-tions, for example in the study of quantum mechanics in harmonic potential. A particularspecial solution of this equation is the parabolic cylinder or Weber function w =: Dα(x),uniquely specified by the asymptotic behaviour

Dα(x) = xα e−x2/4

(1− α(α − 1)

2x2+ O

(1

x4

)), x→ +∞. (2.1.5)

The simplest special case arises when α is an integer n, this yields the Hermite polynomialsHn(x), defined by

Dn(x) = 2−n/2 exp(−x2/4

)Hn(z), z = x/

√2. (2.1.6)

Instead of thinking of Equation (2.1.4) as one equation specified by one fixed value ofα, it is more productive to think of it as an infinite sequence of equations, each of which is

2.1. SPECIAL FUNCTIONS AND LINEAR EQUATIONS 37

specified by successive values of α = n + α0. This alternative perspective has a wonderfulconsequence: we can generate new solutions of each successive equation by knowing solutionsof an earlier equation in the sequence. To see how to do this, note that the differentialoperator in Equation (2.1.4) factorizes:

(∂x − x/2

)(∂x + x/2

)w = −αw. (2.1.7)

Assume that w = wα(x) is a general solution corresponding to a given value of α. Let usdefine

w :=(∂x + x/2

)w. (2.1.8)

Then we have from (2.1.7) the system of two equations

{ (∂x + x/2

)w = w,(

∂x − x/2)w = −αw. (2.1.9)

If we use the first to eliminate w we immediately get (2.1.7), while eliminating w yields

(∂x + x/2

)(∂x − x/2

)w = −α w, (2.1.10)

or after expandingw′′ +

((α− 1) + 1

2 − 14x

2)w = 0. (2.1.11)

which is the same as (2.1.4) except for a new parameter value α = α− 1, and thus wα(x) ∝wα−1(x).

Since the equations are linear the proportionality constant is in principle free, and isdetermined by some other considerations, like normalization. In this particular case thesolutions of (2.1.4) (the parabolic cylinder functions Dα(x)) are conventionally normalizedto have the asymptotic behaviour as given in (2.1.5). Using (2.1.8) on (2.1.5) we get

(∂x + x/2

)Dα(x) = αxα−1 e−x2/4

(1 + O

(1/x2

))= αDα−1,

and therefore we should use the normalization

wα = Dα(x) , wα(x) = αDα−1(x).

Equation (2.1.9) can be written in two additional ways, using this normalization. First bysolving for the term that appears alone:

{Dα−1(x) = 1

α [D′α(x) +

12xDα(x)],

Dα(x) = D′α−1(x)− 1

2xDα−1(x).(2.1.12)

and secondly by solving for the derivative terms:

{D′

α(x) = − 12xDα(x) + αDα−1(x),

D′α−1(x) = 1

2xDα−1(x)−Dα(x).(2.1.13)

The first pair allows us to travel up and down in the chain of Dα(x)’s for different α’sseparated by integers. The second form (2.1.13) allows us to derive a fully discrete equation:


substituting α → α + 1 into the second equation of (2.1.13) and then subtracting the twoequations we get

Dα+1(x)− xDα(x) + αDα−1(x) = 0. (2.1.14)

This is a difference equation, where α is the independent variable and x the parameter.

Let us now reflect on what we have obtained. The conventional theory of ordinarydifferential equations regards the parameters α as given and fixed, while x varies in somedomain. By using the Darboux transformation (2.1.8) we can change the parameter α. Inthe fully dual point of view, we would keep x is fixed while α changes. At an intermediatestep also have recurrence relations that relate w(x;α), w(x; α) and their x-derivatives. Inthe theory of integrable equations, such recurrence relations are called (auto-) Backlundtransformations.

Exercise 2.1.1. Note that we could have factorized the operator in Weber’s equation in adifferent order to get

(∂x + x/2

)(∂x − x/2

)w = −(α + 1)w. Find the equation satisfied by

w = (∂x − x/2)w and deduce another Darboux transformation. Also show that w ∝ wα+1.

Exercise 2.1.2. Use the relationship between Weber function and Hermite polynomials(2.1.6) to derive the difference equation for Hermite polynomials Hn+1(x) = 2xHn(x) −2nHn−1(x), starting from (2.1.14).

Quantum mechanical interpretation: Equation (2.1.4) looks like a Schrodinger equa-tion for a particle moving in a harmonic potential, so the discussion above has a quantummechanical interpretation: ∂x + x/2 is a lowering operator. It kills the lowest energy state:

(∂x + x/2)ψ = 0, from which we find ψ0 ∝ e−x2/4. ∂x − x/2 is the raising operator, whichgenerates excited states starting from the ground state ψ0.

2.1.2 Darboux and Backlund transformation in general

Jean Gaston Darboux (1842 - 1917) was one of the pioneers ofclassical differential geometry. Furthermore, he made importantcontributions to many other fields of mathematics including dif-ferential equations. In his book [Darboux, 1914] (p.210, Section408) a theorem, which has been revived in the modern theoryof integrable systems, leading to what in the modern literature,cf. e.g. [Matveev & Sall, 1991;Rogers & Schief, 2002], is calleda Darboux transformation. Related to it is the notion of a so-called Backlund transformation, which we describe briefly belowand in the next section in more detail (for the case of PDEs).Whereas Darboux’ theorem applies primarily to linear equations,the Backlund transformation, as we shall see, is most relevant tononlinear equations. Figure 2.1: J.G. Darboux

Darboux’ theorem can be described as follows:

Theorem 2.1.1. (Darboux transformation) Let us consider the differential equation

y′′ = [φ(x) + h]y , (2.1.15)


with parameter h. Suppose f(x) is a particular solution of this equation for some specificvalue h1 of h. Let us define a new function y by

y := [∂x − (log f)′]y. (2.1.16)

Then y solves the equation

y′′ = [φ(x) + h]y, where φ := φ− 2(log f)′′. (2.1.17)

We say that φ is the Darboux transform of the potential and y of the wave-function.

What is behind this theorem is a factorisation property. In fact, given that the “poten-tial” function φ(x) does not depend on the (spectral) parameter h, the solution y(x, h1) =f(x) of (2.1.15) allows one to express this potential as

φ(x) = −h1 + ∂2x log f + (∂x log f)2 ,

and hence eq. (2.1.15) for general h can be rewritten as:

y′′ = (h− h1 + v′ + v2)y , where v = ∂x log f .

This can be factorised as follows:

(∂x + v)(∂x − v)y = (h− h1)y ,

and setting y = (∂x − v)y , we obtain by interchanging the factors:

(∂x − v)(∂x + v)y = (h− h1)y ,

the equation (??) with a new potential φ(x) = φ(x) − 2v′ given by (2.1.16).We note that the result on the Weber function of the previous section can be viewed

as an application of Darboux’ theorem: In comparison with (2.1.4) we see that we should

choose φ = x2/4, f = ex2/4, h1 = 1/2. Then we find that φ = φ − 1 in agreement with

(2.1.11).The pair of equations (2.1.12) are a special case of a con-

struction named after Albert Victor Backlund (1845 - 1922) whoworked on transformations of surfaces in differential geometry.In the the modern era the connection between the latter sub-ject and the theory of differential equations has become moreprominent. In fact, there is close relation between transforma-tions between special surfaces in terms of coordinates on thesesurfaces and transformations between solutions of (linear andnonlinear) differential equations. As we shall see in the nextsection these transforms will form the basis of the constructionof exact discretizations of those very same differential equations.We defer the precise definition of a Backlund transformation tolater when we deal with PDEs, but only present here a loose Figure 2.2: A.V. Backlund

definition of this concept as follows:


Definition 2.1.1 (Backlund transformation (loose definition)). Suppose we have a pairof equations depending on two dependent variables u and v and possibly on their partialderivatives: {

F (u, ux, . . . , v, vx, . . . ) = 0,G(u, ux, . . . , v, vx, . . . ) = 0.

(2.1.18)

If upon eliminating v we obtain the equation R(u, ux, uxx . . . ) = 0 and upon eliminating uwe obtain S(v, vx, vxx . . . ) = 0 then (2.1.18) is called a Backlund transformation (BT)between the equations R = 0 and S = 0. If R and S differ only through some parametersthe transformation is called an auto-Backlund transformation (aBT).

We can consider the Darboux transformation y 7→ y described in the theoreom 2.1.1 asa special case of a Backlund transformation. However, we shall mostly employ the term inconnection with transformations between solutions of nonlinear equations, in particular ofnonlinear PDEs and partial difference equations (PδEs).

2.1.3 Another example: Bessel Functions

As an application of the general DT method let us consider the Bessel functions, which alsooccur in many applications, e.g., is the study of waves in circular domains.

Bessel functions are defined as solutions of

x2w′′ + xw′ + (x2 − ν2)w = 0, (2.1.19)

where ν is a parameter, with specific asymptotic behaviours: The standard Bessel functionof the first kind is defined by1

Jν(x) =(x2

)ν ∞∑

k=0

(−x2/4

)k

k! Γ(ν + k + 1). (2.1.20)

We will again consider Bessel’s equation as an infinite sequence of equations in the spaceof the parameter value ν. To allow us to iterate in this parameter space, we need recurrencerelations for Bessel functions. To find Darboux transformations from which recurrencerelations follow, it is easier to start by transforming equation (2.1.19) into a socalled Sturm-Liouville form, i.e. without the first derivative term.

Exercise 2.1.3. Transforming the dependent variable w by setting w(x) = p(x)y(x), andsubstituting this into eq. (2.1.19), show that we can derive a second order ODE for y withouta term containing y′, by choosing 2x p′(x) + p(x) = 0, which implies p(x) = 1/

√x.

Transforming variables by taking w(x; ν) = y(x; ν)/√x, we get

y′′ +

(1− ν2 − 1/4

x2

)y = 0 (2.1.21)

Eq. (2.1.21) corresponds to Darboux’ equation (2.1.15) by setting φ(x) = (ν2−1/4)/x2 andh = −1. Following Darboux’ theorem 2.1.1 we should first construct a suitablefunction f ,

1Recall the definition of the Γ-function, which is used in this formula, as the function obeying the differenceequation Γ(z + 1) = zΓ(z) .


which we can take in the form f := xσ which corresponds to the value h1 = 0 in (2.1.15),and where we need to take σ = (1/2)± ν. Taking, for convenience, the lower sign we obtainthe following transformation from (2.1.17)

y := (∂x + (ν − 1/2)/x) y, (2.1.22)

φ :=ν2 − 1/4

x2− 2(log f)′′ =

(ν − 1)2 − 1/4

x2. (2.1.23)

Thus, we see here again an example here the Darboux transformation implies an integerstep in the parameter, namely the parameter ν, and therefore we can identify

y = yν−1(x), (2.1.24)

up to an arbitrary constant factor, which (as follows from an explicit computation) we canset equal to 1, to be consistent with the series representation (2.1.20) of the solution of theBessel equation.

Using (2.1.22)) we find easily the factorization

(∂x − (ν − 12 )

1x)(∂x + (ν − 1

2 )1x ) = (∂2x − (ν2 − 1

4 )1x2 ) (2.1.25)

and the Backlund pair { [∂x + (ν − 1

2 )/x]y = y,[

∂x − (ν − 12 )/x

]y = − y, (2.1.26)

where eliminating y yields (2.1.21) for y, while eliminating y yields

y′′ +

(1− (ν − 1)2 − 1/4

x2

)y = 0 ,

i.e., eq. (2.1.21) with ν → ν − 1.

Returning to the original variables with y =√xw the above equations become

{ [∂x + ν/x

]w = w,[

∂x − (ν − 1)/x]w = −w. (2.1.27)

Now solving for the derivative terms with (2.1.24) and shifting ν → ν + 1 in the secondequation we obtain {

xw′ν = −ν wν + xwν−1,

xw′ν = ν wν − xwν+1,

(2.1.28)

and after subtracting we get the difference equation in ν with x a parameter:

xwν+1 − 2ν wν + xwν−1 = 0. (2.1.29)

The equations (2.1.28) and (2.1.29) hold for the Bessel functions of the first kind Jν(x).


2.2 Backlund transformations for non-linear PDEs

Many non-linear integrable PDEs possess Backlund transformations and their consistencycondition leads to integrable nonlinear partial difference equations. In this section we willmainly discuss the Korteweg–de Vries (KdV) equation, which is given by the PDE:

ut = uxxx + 6uux . (2.2.30)

This so-called nonlinear evolution equation was derived in1895 by the two people whose names it bears, in a study onshallow water waves and where an exact “solitary wave” solutionwas presented. The equation was a key milestone in a big con-traversy on the nature of waves, not least following the famous“real life” observation of a solitary wave by John Scott Russellin 1834. It is not the place here to recite the whole history ofthe soliton, which can be found in several of the existing mono-graphs on solitons and integrable systems, see e.g. [Ablowitz &Segur, 1982; Calogero & Degasperis, 1982; Newell, 1985; Drazin& Johnson, 1989; Ablowitz & Clarkson, 1991]. We just mentionthat the KdV equation was revived 70 years after Korteweg andde Vries’ paper, when in a celebrated study by C. Gardner, J. Figure 2.3: D. Korteweg

Greene, M. Kruskal and R. Miura, it was shown that thenonlinear PDE (2.2.30) can beexactly solved by an ingenious method, which is nowadays referred to as the inverse scat-tering transform method. Although this method is only applicable to very special equations,equations that we refer to as soliton equations or exactly integrable equations, we now knowentire infinite families of such equations to which the method can be applied to find exactsolutions of the nonlinear equations (this being in stark contrast with the generic situationthat nonlinear PDEs in the general case cannot be exactly solved and that typically we haveto resort to either qualitative studies or perturbative and numerical methods to study theirsolutions).

2.2.1 Lax pair for KdV

One of the key properties of this equation is that there exists an underlying overdeterminedsystem of linear equations:

ψxx + uψ = λψ , (2.2.31a)

ψt = 4ψxxx + 6uψx + 3uxψ, (2.2.31b)

whose consistency condition leads to (2.2.30). The first equation (2.2.31a) has the form ofa linear spectral problem for the differential operator

L = ∂2x + u , (2.2.32)

where the coefficient u = u(x, t) plays the role of a potential. The parameter λ is aneigenvalue of the operator L and can, in principle, depend on t if u depends on it, butby the definition of an eigenvalue of a differential operator it should independent of x. The

2.2. BACKLUND TRANSFORMATIONS FOR NON-LINEAR PDES 43

second equation (2.2.31b) describes the (linear) time-evolution of the function ψ(x, t), whereagain the same u enters in the coefficients. The system (2.2.31) is overdetermined: the twolinear equations can only be compatible with each other if additional conditions hold for thecoefficients, which are all expressed in terms of u. Furthermore, we shall assume that thetime-evolution descibes an isospectral deformation2 of the linear spectral problem, i.e., weassume that λ is, in fact, independent of t.

Theorem 2.2.1. Under the assumption of isospectrality, i.e., λt = 0 , the linear system(2.2.31) is self-consistent, i.e., (ψxx)t = (ψt)xx , iff either ψ ≡ 0 or the potential u = u(x, t)obeys the Korteweg-de Vries (KdV) equation (2.2.30).

Proof. First, using (2.2.31a) rewrite (2.2.31b) as follows:

ψt = (4λ+ 2u)ψx − uxψ .

and now consider the cross-derivatives:

(ψxx)t = ((λ− u)ψ)t = (λt − ut)ψ + (λ− u) [(4λ+ 2u)ψx − uxψ](ψt)xx = (4λ+ 2u)ψxxx + 4uxψxx + 2uxxψx − uxψxx − 2uxxψx − uxxxψ

= (4λ+ 2u) ((λ− u)ψ)x + 3ux(λ− u)ψ − uxxxψ⇒ (ψt)xx − (ψxx)t = (ut − uxxx − 6uux)ψ − λtψ .

Hence, under the condition of isospectrality, λt = 0 , we see that the system is compatible,i.e. (ψt)xx = (ψxx)t provided u(x, t) obeys the KdV equation.

The linear system of equations associated with a non-linear PDE is symptomatic of itsintegrability through the inverse scattering method. Such a linear system, consisting of aspectral problem and an equation for the time evolution, is called a Lax pair, after P.D.Lax who gave a systematic framework for describing such linear problems in his celebratedpaper [Lax,1968]. Although for virtually all soliton equations Lax pairs have been found,there is no fully algorithmic method known to produce a Lax pair for a given equation.

2.2.2 Miura transformation

The KdV equation possesses a remarkable transformation, called the Miura transformationafter its inventor, which gives rise to many insights about its solutions, and in particularcan be used to derive a Backlund transformation for KdV. To find it, consider how wecan eliminate the function u from the system (2.2.31) and obtain a PDE in terms of the“eigenfunction” ψ itself.

From (2.2.31a) we findu = λ− ψxx/ψ,

and inserting this into (2.2.31b) we obtain the following equation for ψ

ψt = ψxxx − 3ψxψxx

ψ+ 6λψx . (2.2.33)

2By ”deformation” we understand the variation of the operator L in (2.2.32) through the change of thepotential u(x, t) as t varies, which in principle may affect the spectrum of the operator, i.e., the collection ofeigenvalues. However, if the deformation is ”isospectral” the spectrum is preserved while varying t, meaningthat the eigenvalues λ do not change as t varies.


Introducing the variable:v := ∂x log ψ , (2.2.34)

we easily obtain from (2.2.31a) the Miura transformation

u = λ− vx − v2 . (2.2.35)

Furthermore, after taking derivatives w.r.t. x on both sides of (4.2.29) and expressing allterms using v we get a PDE governing v:

vt = vxxx − 6v2vx + 6λvx . (2.2.36)

The latter equation (for λ = 0) is known as the modified KdV equation (MKdV), andit differs from the KdV equation (2.2.30) notably in the nonlinear term. The differentialsubstitution (2.2.35) allows one to find a solution of the KdV equation given a solution ofthe MKdV equation: if v solves the MKdV (2.2.36) and u is defined by (2.2.35), then usolves KdV (2.2.30).

2.2.3 The Backlund transformation

Let us now turn to the derivation of Backlund transformations using the above. We startwith the Miura transformation (2.2.35), and combine it with the simple observation thatthe equation (2.2.36) is invariant under the replacement v 7→ −v . The idea is to use onesign in transforming from u to v and another in transforming from v to u, that is, we willhave the Miura transformations

u = λ+ vx − v2 , (2.2.37a)

u = λ− vx − v2 . (2.2.37b)

It is surprising that the trivial transformation v 7→ −v implies a highly nontrivial trans-formation u 7→ u on the solutions of the KdV equation.

Adding and subtracting the two relations above we obtain

u+ u = 2(λ− v2) (2.2.38a)

u− u = 2vx. (2.2.38b)

The latter can be integrated if we introduce the variable w by taking u = wx. For the KdVequation this change of variables leads to

wt = wxxx + 3w2x, (2.2.39)

after one integration in x. (Note that we have omitted an irrelevant integration constant.)This equation for w is called the potential KdV equation (PKdV).

In terms of this new dependent variable the equation (2.2.38b) can be integrated tow − w = 2v , and inserting it into the first relation (2.2.38a) we obtain

(w + w)x = 2λ− 1

2(w − w)2 . (2.2.40a)


written entirely in terms of w.Equation (2.2.41) provides us with the x-dependent part of the Backlund transformation

. To fully characterize the solution w we need also a t-dependent equation, which can bereadily found by using the PKdV equation itself. Adding (2.2.39) for w and w and using(2.2.41) to reduce wxxx + wxxx we obtain the relation

(w + w)t = (w − w) (wxx − wxx) + 2(w2x + wxwx + w2

x) , (2.2.40b)

and the relations (2.2.41), (2.2.42) together constitute the Backlund transformation for theKdV equation. (Note that in practice, one could use the PKdV equation itself rather than(2.2.42) to implement the BT.)

Supplementing the latter relation by a relation for the t derivatives, using the PKdVitself, we obtain the following statement:

Theorem 2.2.2. The system of relations

(w + w)x = 2λ− 1

2(w − w)2 , (2.2.41)

(w + w)t = (w − w) (wxx − wxx) + 2(w2x + wxwx + w2

x) , (2.2.42)

defines a transformation from a given solution w(x, t) of the PKdV to a new solution w(x, t)of the PKdV equation.

Proof. Differentiating (2.2.41) by t and (2.2.42) by x we get respectively:

(w +w)xt = −(w − w)(wt − wt)

(w +w)tx = (wx −wx)(wxx − wxx) + (w − w)(wxxx − wxxx)

+2 (2wxwxx + 2wxwxx + wxwxx + wxwxx)

= (w −w)(wxxx − wxxx) + 3(wx + wx)(wxx + wxx)

and subtracting the first from the second we get, using also the x-derivative of (2.2.41),

0 = (w − w) [(wt − wxxx)− (wt − wxxx)] + 3(wx + wx) [−(w − w)(wx − wx)]

= (w − w)[(wt − wxxx − 3w2

x)− (wt −wxxx − 3w2x)],

and hence, if w solves the PKdV equation then either w = w or w solves the PKdV as well.

2.2.4 Using BTs to generate multisoliton solutions

Note that the Backlund pair (2.2.40) is different from the one we had before in that it containsa parameter that does not appear at all in its base equation (2.2.39). This parameter canbe used to generate more complicated solutions from simpler ones.

Suppose we know a given “seed solution” solution w of the PKdV, then inserting thisinto (2.2.41) we obtain a first order nonlinear ODE for w. This ODE will always be of theform:

wx = −1

2w2 + a(x)w + b(x),

where the right-hand side is a quadratic in w (with x-dependent coefficients). This is awell-known type of differential equation called a Riccati equation. These equations aregenerally solvable through a linearisation procedure. After solving this equation we havesome integration constants that may depend on t, they can be determined from (2.2.42).


Example: As a specific example, consider the simplest case where the seed solution w of thePKdV equation (2.2.39) is the trivial solution w ≡ 0. Setting w ≡ 0 in (2.2.41) yields

wx = 2λ − 1

2w2,

which can be integrated by separation of variables and yields

w(x, t) = 2k tanh (kx+ c(t)) , λ = k2.

Substituting this expression into the (2.2.42) (with w = 0) reveals that we must take ct = 4k3t+ c0,were c0 is a constant. Thus we have obtained the solution:

w(x, t) = 2k tanh(kx+ 4k3t+ c0

). (2.2.43)

This implies that for the solution of the KdV equation we obtain the solution

u(x, t) = wx = 2k2 sech2 (kx+ 4k3t+ c0), (2.2.44)

which is the famous formula for the 1-soliton solution of the KdV equation.

2.2.5 Permutability property of BTs

The solution we obtained above can now be regarded as the starting point for applyingthe BT once again to obtain yet another solution of the PKdV equation. Carrying this outfurther we can iteratively obtain an infinite sequence of increasingly complicated solutions ofthe same nonlinear PDE. The procedure of solving a Riccati equation at each stage obviouslybecomes increasingly more cumbersome as we go along. However, there is a powerful newingredient that can be used to simplify the iteration, namely the permutability property ofthe BTs.

Suppose we want to compose two different BTs, one with a parameter λ, as in (2.2.40),and one with another parameter, say µ, given by

BTλ : wλ7→ w (w + w)x = 2λ− 1

2(w − w)2 , (2.2.45a)

BTµ : wµ7→ w (w + w)x = 2µ− 1

2(w − w)2 , (2.2.45b)

where we have used the notation w, w to denote the solution obtained by applying the BTwith parameter λ, µ, respectively.

There are now two ways to compose these BTs: either start with BTλ and and subse-quently apply BTµ, or the other way around. In this way we get iterated solutions which

we can denote by w and ˜w respectively,

w = BTµ ◦BTλw, ˜w = BTλ ◦BTµw.

The highly nontrivial result is that, under certain conditions, both ways of composing BTs

lead to the same result: w = ˜w, and hence the two BTs commute. This is the famouspermutability property of the BTs.


Theorem 2.2.3. The BTs given by (2.2.45a),(2.2.45b) for different parameters λ and µgenerate solutions (with a suitable choice of integration constants) for which we have thefollowing commutation diagram of BTs:

w

w

w

BTλ

BTµ

w = ˜w

BTµ

BTλ

The proof of the permutability property is quite deep and relies on the spectral propertiesthat play at the background of the equations.

Proof. A direct proof is computational. Using the additional relations:

BTµ : wµ7→ w

(w + w

)x= 2µ− 1

2

(w − w

)2, (2.2.45c)

BTλ : wλ7→ ˜w

(˜w + w

)x= 2λ− 1

2

(˜w − w

)2. (2.2.45d)

Iterating the first Backlund chain we can solve w in terms of w and w as follows:

w =1

2( w + w) +

( w − w)x + 2(λ− µ)w − w

,

and reinserting this into the BT we obtain:

λ+ µ = ( w + w)x + ∂2x log(w − w) + 1

2

(∂x log( w − w)

)2

+1

8( w − w)2 + 2

(λ− µ)2

( w − w)2,

and the latter relation is symmetric under the interchange of λ and µ. A similar symmetrycan be derived for the t-part of the BT. Hence, starting from an arbitrary seed w we canfind solutions, by appropriate choice of integration constants, which are symmetric underinterchange of λ and µ.

The consequences of the permutability property are far reaching, and we will give anexplicit realization as follows. In fact, using all four Backlund relations (2.2.45a)-(2.2.45d),

and setting w = ˜w, we can now eliminate all the derivatives from the four eqs. (2.2.45) weobtain a purely algebraic equation of the form:

( w − w)(w − w) = 4(µ− λ). (2.2.46)

This allows us to obtain directly the iterated BT transformed variable w without having toderive the solution through the Riccati equations of the BT.


Exercise 2.2.1. Construct a 2-soliton solution by starting from the seed solution w ≡ 0,and two 1-soliton solutions of the type (2.2.43) (with different parameters k and l, where

λ = k2, µ = l2, respectively) and solving for w from (2.2.46). Note that the phases c0 may

be taken to be different for w and w. Verify explicitly that the w so constructed actuallysolves (2.2.39).

2.2.6 Backlund transformation for the sine-Gordon equation

BTs exist for other integrable evolution equations as well. In fact the first BT, the oneproposed by Backlund himself, is associated with the sine-Gordon equation,

θxt = sin θ , (2.2.47)

For this equation the Backlund transformation θλ→ θ is given by the following relations:

(θ − θ

)x

= 2λ sin

(θ + θ

2

), (2.2.48a)

(θ + θ

)t

=2

λsin

(θ − θ2

), (2.2.48b)

connecting a variable θ(x, t) to a new variable θ(x, t). By calculating the t derivative (2.2.48a)and the x-derivative (2.2.48b) and then taking a sum or difference, one can easily derive

(2.2.47) for θ or θ, respectively. Thus (2.2.48) is a one-parameter auto-Backlund transfor-mation for the sine-Gordon equation.

As before, we can now introduce a second BT θµ→ θ of the form (2.2.48) with parameter

µ, namely

(θ − θ

)x

= 2µ sin

(θ + θ

2

), (2.2.49a)

(θ + θ

)t

=2

µsin

(θ − θ2

). (2.2.49b)

We can also apply BTµ on θ, and BTλ on θ, and if among these 8 equations we eliminate all

derivatives (under the assumption of the permutability of the BTs, i.e.,θ =

˜θ ), we obtain

the following permutability property:

sin

θ + θ − θ − θ

4

=

λ

µsin

θ + θ − θ − θ

4

. (2.2.50)

If, for simplification, we denote ei2 θ = w, we can write (2.2.50) as

λ( ww − ww) = µ( ww − ww). (2.2.51)


w

w

w

w

w

˜w

˜w

w

w

w

Figure 2.4: A lattice of BTs

Exercise 2.2.2. Starting from the trivial solution of the sine-Gordon equation θ ≡ 0, usethe BT (2.2.48) to obtain a solution θ(x, t) of the same equation containing the parameterλ, namely:

θ(x, t) = 4 tan−1

{exp

(λx +

t

λ+ ϕ0

)}, (2.2.52)

where ϕ0 is a (constant) phase. (Hint: Use the integral∫

dϕ

sinϕ= ln

(tan

ϕ

2

)+ c.

2.2.7 Transition to lattice equations

Is obvious from the above, that by iterating the BTs with two different parameters we obtainfrom one seed solution w an entire lattice of solutions, see Figure 2.4. Note that buildingthis lattice of solutions crucially depends on the validity of the permutability property!

We have derived the permutability equations (2.2.46) and (2.2.51) from the propertiesof the PKdV and SG, respectively, thus these equations are descriptive, they describe yetanother property of the sequence of functions derived using BTs. We can introduce anenumeration of the solutions as follows:

wn,m = BT nλ ◦BTm

µ w, (2.2.53)

after which we can write (2.2.46) and (2.2.51) as difference equations of the form:

(wn+1,m+1 − wn,m)(wn,m+1 − wn+1,m) = 4(µ− λ), (2.2.54)

and

λ(wn+1,m+1wn,m+1 − wn+1,mwn,m) = µ(wn+1,m+1wn+1,m − wn,m+1wn,m), (2.2.55)


respectively, the shifts along the lattice wn,m 7→ wn+1,m and wn,m 7→ wn,m+1 correspondingto the application of the Backlund transformations BTλ and BTµ.

We will now change our point of view: at each elementary plaquette of the above latticeof solutions we have a relation of the form (2.2.54) or (2.2.55) (or something else for otherequations), and we will now elevate these equations as being the main equations of interest.In a sense having reached this point we can “forget” about the original PDE’s from whencethe construction originated, and place the permutability equations at the centre of our focus.

Later we will indeed consider lattice equations, i.e., P∆E’s like (2.2.54) or (2.2.55) ontheir own merit and study their remarkable properties. In particular we will show that theseequations are integrable in some precise sense.

Literature:

1. E.T. Whittaker and G.N. Watson, A course of Modern Analysis, fourth edition, 1927,(Cambridge University Press, 2002).

2. J.G. Darboux Lecons sur la theorie generale des surfaces et les applications geometriquesdu calcul infinitesimal (Gauthier-Villars, Paris, 1914)

3. V.B. Matveev and M.A. Salle, Darboux transformations and solitons (Springer, 1991)

4. C. Rogers and W.K. Schief, Backlund and Darboux Transformations (Cambridge Uni-versity Press, 2002).

5. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in arectangular channel, and on a new type of long stationary waves, Philos. Mag. (5) 39(1895) 422–443.

6. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving theKorteweg-de Vries equation, Phys. Rev. Lett. 19 (1967) 1095–1097.

7. P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun.Math. Phys. 21 (1968) 467–490.

8. R.M. Miura, The Korteweg-de Vries Equation: A Survey of Results, SIAM Review 18# 3 (1976) 412–459.

9. G.L. Lamb, Elements of Soliton Theory, (Wiley Interscience, 1980).

10. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM,1982).

11. F. Calogero and A. Degasperis, Spectral Transform and Solitons, vol. 1, (North-Holland Publ., Amsterdam, 1982).

12. A.C. Newell, Solitons in Mathematics and Physics, (SIAM, Philadelphia, 1985).

13. P.G. Drazin and R.S. Johnson, Solitons: An Introduction, (Cambridge UniversityPress, 1989).

14. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and InverseScattering, (Cambridge Univ. Press, 1991).

Lecture 3

Integrability of P∆Es

Motivated by the lattice structure emerging from the permutability/superposition propertiesof the Backlund transformations of the previous Lecture, we will now consider the integrabil-ity properties of these viewed as partial difference equations (P∆Es) on the two-dimensionalspace-time lattice. What we will discover is that the integrability can be given a precise,even algorithmic, meaning. The presence of parameters (namely the Backlund parametersλ and µ, which we will now reinterpret as lattice parameters) will play a crucial role inthe development of the theory. Furthermore, as we will see in subsequent Lectures, theparameters render the P∆Es very rich: since they can be seen to represent the widths of theunderlying lattice grid they allow us to recover, through continuum limits, a great wealthof other equations, semi-continuous (i.e. differential-difference type) as well as fully contin-uous (i.e. partial differential type) ones. The interplay between the discrete and continuousstructures will prove to be one of the emerging features of the integrable systems that westudy.

The history of integrable difference equations goes back to seminal papers by Ablowitz& Ladik and by Hirota in the 1970’s, cf. [1,2]. The first was motivated by the searchfor integrable numerical algorithms through finite-difference approximations. This fits wellinto the general problem of the analysis of finite-difference P∆Es arising from numericalstudies of PDEs, cf. e.g. the monograph by P.R. Garabedian, Ch. 13. More recently,systematic methods for the construction of integrable nonlinear finite-difference P∆Es werefound, e.g. through the representation theory of infinite-dimensional Lie algebras, [3], orthrough singular linear integral equations and connections with Backlund transformations,cf. [4,5].

3.1 Quadrilateral P∆Es

We will investigate here partial difference equations (P∆E’s) of the following canonical form(which we will call quadrilateral P∆Es)

Q(u, u, u, u) = 0 , (3.1.1)

51

52 LECTURE 3. INTEGRABILITY OF P∆ES

where we adopt the canonical notation of vertices surrounding an elementary plaquette ona rectangular lattice:

u := un,m , u = un+1,m

u := un,m+1 , u = un+1,m+1

Schematically, this configuration of points is given by:

- -

?

?

- -

?

?

uu

uu

The notation is inspired by the one for Backlund transformations, which as we haveseen in the previous Lecture, give rise to purely algebraic equation as a consequence of theirpermutability property. However, here we will forget, at first instance, about this connection,and consider lattice equations of the form (3.1.1) in their own right as a partial differenceequation (P∆E) on a two-dimensional lattice. Thus, n and m, play the role of independent(discrete) variables, very much like x and t being the continuous variables for an equationlike the KdV equation

Even though the form (3.1.1) seems very restrictive, it will turn out that it is in a sense themost elementary form as a model type of equations, and on the other hand remarkably rich.In fact, as we shall see later, P∆Es of the form (3.1.1) are “rich enough” to be candidatesfor discretisations of PDEs of arbitrary order in the spatial and temporal variables.

A classification of P∆Es in the same way as of PDE s, does not yet exist. Nonetheless, wemay consider a P∆E of the form (3.1.1) to have features reminiscent of hyperbolic PDEs.In fact, if the equation Q = 0 can be solved uniquely for each elementary quadrilateral,then one may pose initial value problems (IVPs) in ways very similar to hyperbolic type ofequations such as the KdV equation itself (i.e. as a discrete nonlinear evolution equation).This can be done as follows.

The naive approach would be to consider IVP where we would assign values of thedependenet variable u along horizontal array of vertices in the lattice, i.e. values for un,0,for all n, considering the variable m to be the temporal discrete variable. It is easy to see,however, that such an IVP would lead to a nonlocal problem if we want to use (3.1.1) as aniteration scheme to find all values un,m for m > 0. In fact, to calculate any value un,1 sayfor given n, we would need to involve all initial values un′,0 with n′ < n, and furthermorehave to assume limiting behaviour as n→ −∞. This would be a complicated procedure.

However, there is nothing that tells us that we should identify the n- and m axes as thespatial and temporal axes respectively. The lattice picture allows us to play other and more

3.1. QUADRILATERAL P∆ES 53

natural games, and it is the equation itself that gives the lead in provding is with the naturalIVP that we could impose. Thus, changing the perspective slightly, we may tilt the latticeand rather consider initial value data to be imposed on configurations like a “sawtooth” (ora “ladder”), such as:

R

�

R

�

R

�

The analogy with hyperbolic PDEs can be further seen from the consideration of the“memory” each point in the lattice has of the initial values that are involved in its determi-nation upon iteration. In fact, any given point has a backward shadow (the analogue of theso-called “lightcone”) of points the values of u on which determine the value at that point,as indicated by the picture:

the point at the bottom of the cone being fully determined by the initial values at the topof the cone and only and exclusively by those values! This is very much reminiscent againof what happens in the case of hyperbolic PDEs.

Multilinearity In order to have a unique iteration scheme arising from an equation ofthe form (3.1.1) the equation should be linear in each of the variables around the quadrilat-eral, i.e. the equation should multilinear. The most general form of a quadrilateral latticeequation which is linear in each dependent variable around the quadrilateral, and which inaddition respects reversal symmetry with respect to the shifts ˜- and on the lattice, takesthe following form:

k0uuuu+ k1

(uuu+ uuu+ uuu+ uuu

)+ k2

(uu+ uu

)

+k3

(uu+ uu

)+ k4

(uu+ uu

)+ k5

(u+ u+ u+ u

)+ k6 = 0 , (3.1.2)

where k1, . . . , k6 are coefficients (which may depend on additional parameters).

Exercise 3.1.1. Show that, starting from a multilinear quadrilateral lattice equation (3.1.1)with 16 general coefficients, we arrive at the form (3.1.2) by assuming that the equationremains unchanged when we reverse the ˜- or shifts, i.e. when we replace u by

˜u (by which

we mean the backward shift related to the ˜-shift, see picture), or when we replace u byu

(meaning the backward shift related to the -shift, see picture).


˜u u u

u

u

u

It is easy to see that the lattice equation (2.2.54) derived in Lecture 2 from the Backlundtransform of KdV is precisely of the form given above. However, not all equations of theform (3.1.2) will be of interest to us. We will be interested in those equations (for specificchoices of the coefficients k1, . . . , k6) which have the additional property of being integrable.What we mean by this will be explored in the next section.

3.2 Integrability and Consistency-around-the-Cube

We will now consider a class of quadrilateral P∆Es (3.1.1) in which, apart from the inde-pendent discrete variables n,m on which the variable un,m depends, there are parameterswhich we associate with these independent variables. We can think of these parameters asbeing the parameters which measure the width of the grid in the directions associated withn and m, and we refer to them as lattice parameters. Thus, denoting these parameters byp, q, the equations under question will take the form:

Q(u, u, u u; p, q) = 0 , (3.2.1)

and if we demand that the values for u at each vertex can be solved uniquely, implyingmultilinearity in each of these variables around the quadrilateral, then we are led again toeq. (3.1.2) with the coefficients k0, . . . , k6 depending on the lattice parameters p and q in aspecific way. The question is what criteria to use in order to chose that dependence!

It is here that we will restrict ourselves to quadrilateral P∆Es which we regard to beintegrable. The question of what is the proper definition integrability, and to answer thatquestion in general is very difficult: a one-fits-all definition (for all the possible type ofsystems that we would like to regard as being integrable) is possibly not possible to givein a precise mathematical sense. However, if we restrict ourselves here to P∆Es, and inparticular quadrilateral P∆Es, then we can aspire to be a bit more precise. We will explorethe definition of an integrable quadrilateral P∆E by means of the following example, namelythe example of the P∆E arising from the BTs for the KdV equation, eq. (2.2.54).

First, we remark that the presence of the lattice parameters p, q is crucial: whereasnormally we would like to consider the parameters to be chosen once and for all and thenremain fixed (thus specifying a specifi equation) when solving the equation on the lattice,here we will argue that we should look at (2.2.54) as defining a whole parameter-family ofequations, and that it makes sense to look at them altogether with p and q variable. In

3.2. INTEGRABILITY AND CONSISTENCY-AROUND-THE-CUBE 55

doing this, however, we must attach each parameter to a specific diecrete variable such asp being associated with the variable n, and q with m. This is also natural from the way inwhich we derived (2.2.54) from the BT construction: each BT is attached to a parameter (λ,µ,. . . ), and with each parameter we can build a direction in an infinite-dimensional latticeof BTs.

The main point we want to make now is the following:

Statement: the infinite parameter-family of P∆Es represented by the lattice equation,such as (2.2.54) is compatible, i.e. in each quadrilateral sublattice of the infinite-dimensionallattice we can consistently impose a copy of the P∆E in terms of the relevant discretevariable and associated with a corresponding lattice parameter.

Let us illustrate this by means of the example of (2.2.54), in which we will identify (forlater convenience) the parameters 4λ := p2 and 4µ := q2. What the statement abovesuggests is that we can “embed” the equation in a multi-dimensional lattice by conbsideringthe dependent variable w, not only to depend on n and m (with associated lattice paametersp and q espectively), but that w may in fact depend on an infinity of lattice variables eachassociated with its parameter, as follows:

w = wn,m,h,... = w(n,m, h, . . . ; p, q, r, . . . )

and with each of these variables we have a corresponding elementary shift on the lattice:

w := wn+1,m,h , w := wn,m+1,h , w := wn,m,h+1 . . .

Rewriting the equation (2.2.54) in the form (2.2.46) with the substitutions for λ and µinvestigate what happens if we impose a copy of the same equation in all three latticedirections. This would lead to the system of equations:

(w − w)(w − w) = p2 − q2 , (3.2.2a)

(w − w)(w − w) = p2 − r2 , (3.2.2b)

(w − w)(w − w) = r2 − q2 . (3.2.2c)

These equations are consistent if the evaluations along the cube are independent of the wayof calculating the final point. Imposing initial values:

w := a , w =: b , w =: c , w =: d


w

w w

w

w

w

w

w

In fact:w =

(p2 − q2)ww + (q2 − r2)ww + (r2 − p2)ww(r2 − q2)w + (q2 − p2)w + (p2 − r2)w . (3.2.3)

independent of the way in which he value at this vertex is calculated! This property, implyingthat under relevant initial value problems on the 3-dimensional lattice the iteration of thesolution can be performed in an unambiguous way (namely independent of the way by whichwe perform the calculation of the iterates) will be referred to as the consistency-around-the-cube (CAC) property. It is this property that we shall consider to be the main hallmarfk ofthe integrability of the equation.

Remark: We note also, for later reference, that in the above case the formula for w isindependent of the value w at the opposite end of the main diagonal across the cube. Thisproperty we will refer to as the tetrahedron property.

Other examples of integrable quadrilateral P∆Es: There are many examples ofintegrable quadrilateral P∆Es that have been discovered over the years. A number of thesebelong to the lattice KdV-family of equations, they comprise the following cases:

Lattice potential KdV equation:

(p− q + u− u)(p+ q + u− u) = p2 − q2 , (3.2.4)

and this is, in fact, equivalent to the equation (2.2.54) by the change of (dependent)variable w = u− np−mq − c (c a constant with respect to n and m). We will showin Lecture 4 that this equation reduces to the potential KdV equation after a doublecontinuum limit, so that we rightly regard it as a discretisation of the latter equation.

Lattice potential MKdV equation:

p(vv − vv) = q(vv − vv) (3.2.5)

Solutions of this equation are related to the solutions of the previous one (3.2.4) viathe relations

p− q + u− u =pv − qv

v, p+ q + u− u =

pv + qvv

, (3.2.6)

3.3. LAX PAIR FOR LATTICE POTENTIAL KDV 57

which constitute the analogues of the Miura transformation (2.2.35).

Lattice SKdV equation:

(z − z)(z − z)(z − z)(z − z)

=p2

q2(3.2.7)

which is the “Schwarzian KdV equation”. This equation is invariant under Mobiustransformations:

z 7→ Z =αz + β

γz + δ.

Solutions of eq. (3.2.7) are related to the ones of (3.2.5) via the relations:

p(z − z) = vv , q(z − z) = vv . (3.2.8)

Exercise 3.2.1. Show by explicit computation that the P∆Es given by (3.2.4)-(3.2.7) possessthe consistency-around-the-cube property.

Exercise 3.2.2. Show that by using the relations (3.2.6) one can derive (3.2.4) from (3.2.5)and vice versa.

3.3 Lax pair for lattice potential KdV

We shall next demonstrate how the CAC property explained in the previous section gives riseto the existence of a Lax pair, i.e. an overdetenrmined linear system of difference equations,the compatibility of which is verified iff the nonlinear lattice equation is satisfied. The ideais the following: Having verified the consistency of the equation around the cube, this tellsus that we can add any lattice direction to the original lattice and impose simultaneouslythe equation in the three two-dimensional quadrilateral sublattices. The main idea is nowto consider the additional lattice variable h ∈ Z associated with lattice parameter k asan auxiliary “virtual” variable, whilst acknowledging only the shifts in the orignal latticevariables n and m as the operations of interest. This then suggests that the shift in thethird direction: w 7→ w should not appear in any of the equations, implying that whereverw appears we should treat it as a new dependent variable w :=W .

Proceeding in this way, we rewrite (3.2.2b) and (3.2.2c) as follows:

(W − w)(W − w) = k2 − p2 ⇒ W =wW + (k2 − p2 − ww)

W − w , (3.3.1a)

(W − w)(W − w) = k2 − q2 ⇒ W =wW + (k2 − p2 − ww)

W − w . (3.3.1b)

Noting that eqs. (3.3.1) are both fractional linear in W , we can linearise these equations bythe substitution:

W =F

G,


leading to

F

G=

wF + (k2 − p2 − ww)GF − wG ,

F

G=

wF + (k2 − q2 − ww)GF − wG ,

and since at least one of the two functions F or G can be chosen freely, this allows us tosplit in each of these equations the numerator and denominator to give:

{F = γ

(wF + [k2 − p2 − ww]G

),

G = γ (F − wG) , respectively

{F = γ′

(wF + [k2 − q2 − ww]G

),

G = γ′ (F − wG) ,(3.3.2)

in which γ and γ′ are to be specified later. What happens next is obvious: we introduce thetwo-component vector

φ =

(FG

),

and write (3.3.2) as a system of two 2×2 matrix equations:

φ = Lφ , φ = Mφ , (3.3.3a)

with the matrices

L = γ

(w k2 − p2 − ww1 −w

), M = γ′

(w k2 − q2 − ww1 −w

)(3.3.3b)

How does this linear system work? The consistency relation of the linear problem (3.3.3a)

is obtained from the condition thatφ =

˜φ , which is the condition that simply expresses

that φ must be a proper function of the lattice variables n and m. Calculating the lefthand-side of this condition we get on the one hand

φ = (Lφ) = Lφ = LMφ ,

whereas the right hand-side would be calculated as

˜φ = (Mφ) = Mφ = MLφ .

Equating both sides we see that a sufficient condition for the consistency is the matricialequation:

LM = ML , (3.3.4)

which we will loosely refer to as the Lax equation, but it is sometimes also referred to asa discrete zero-curvature condition (the reason for this terminology will be explained in(3.3.1)). Pictorially the Lax equation is illustrated by the following diagram

3.3. LAX PAIR FOR LATTICE POTENTIAL KDV 59

φ φ

φφ

L

MM

L

where the vectors φ are located at the vertices of the quadrilateral and in which the matricesL and M are attached to the edges linking the vertices.

Using the explicit form (3.3.3b) of the matrices L and M , and working out the condition(3.3.4) we find

γγ′

(w k2 − p2 − w w1 − w

) (w k2 − q2 − ww1 −w

)=

= γ′γ

(w k2 − q2 − w w1 − w

) (w k2 − p2 − ww1 −w

)

and we will chose the γ and γ′ (which were unspecified so far) such that the relation for thedeterminants of this equation:

γγ′ det(L) det(M) = γ′γ det(M ) det(L) (3.3.5)

is trivially satisified. Since in this example the determinants of L and M are given by

det(L) = p2 − k2 , det(M) = q2 − k2 ,

respectively, it is clear that the condition (3.3.5) is satisified by simply taking γ = γ′ = 1.But we will encounter other examples where a nontrivial choice of γ and γ′ is needed inorder to satisfy the determinantal condition (3.3.5). Working out all the entries on bothsides of the above matrix products it is straightforward to see that the (1,1)- and (2,2) entryof the matrix equation both yield the same condition on w, namely the equation (2.2.46)!Moreover, the (2,1) entry is trivially, and the (1,2) entry we don’t even need to calculate,because having checked three of the entries to be satisfied the final entry must also besatisfied by virtue of the fact that we have aranged the determinantal condition (3.3.5) tobe satisfied. In conclusion, we see thus that from the Lax equation (3.3.4) we recover thelattice equation (2.2.54). We observe furthermore, that albeit both Lax matrices L and M

depend on the auxiliary variable k the final nonlinear equation for w does not depend on k!

Exercise 3.3.1. Suppose that the Lax matrices L and M can be expanded in a power seriesin a small parameter δ and ǫ respectively as follows:

L = 1+ δL1 + · · · , M = 1+ ǫM1 + · · · ,


and we expand the shifted variable by Taylor expansion

L = L+ ǫ∂tL+ · · · , M = L+ δ∂xM + · · · ,

by expansion we obtain in dominant order (namely terms proportional to ǫδ) the followingmatricial equation:

∂tL1 − ∂xM 1 + [L1 , M1 ] = 0 , (3.3.6)

where [L , M ] = LM −ML denotes the usual matrix commutator bracket. Eq. (3.3.6)arises also in differential geometry and it is from there that it has the interpretation asa zero-curvature condition of differential manifolds (with the relevant interpretation of thematrices L and M).

3.4 ∗ Classification of Quadrilateral P∆Es

In a beautiful paper by V. Adler, A. Bobenko and Yu. Suris, cf. ref. [8], the problem ofclassifying all quadrilateral lattices which are integrable in the sense of the CAC propertydiscussed in section 3.2 was considered. Since this is in our view an important result, it isuseful to reproduce the whole list of resulting equations here.

Theorem ([8]): Consider quadrilateral P∆Es of the general form:

Q(u, u, u, u; p, q) = 0 ,

using the notation indicated in the following diagram (renaming the lattice parameters inorder to avoid confusion with previously used notation),

- -

?

?

- -

?

?u u

uu

p

p

q q

subject to the following restrictions:a) Linearity: Q is multilinear in its arguments, i.e., it is linear in each vertex-variable u,u,

u,u;b) Symmetry: Q is invariant under the group D4 of symmetries of the square, generated bythe interchangements:

Q(u, u, u, u; p, q) = ±Q(u, u, u, u; q, p) = ±Q(u, u, u, u; p, q) ; (3.4.1)

c) Tetrahedral Condition: in the consistency check, the evaluation of the point on the cube

given by u is independent of u.

3.4. ∗ CLASSIFICATION OF QUADRILATERAL P∆ES 61

Q-list:

Q1 : p(u− u)(u − u)− q(u − u)(u − u) = δ2pq(q − p) (3.4.2a)

Q2 : p(u− u)(u − u)− q(u − u)(u − u) + pq(p− q)(u + u+ u+ u) == pq(p− q)(p2 − pq + q2) (3.4.2b)

Q3 : p(1− q2)(uu+ uu)− q(1− p2)(uu+ uu) =

= (p2 − q2)((uu+ uu) + δ2

(1 − p2)(1− q2)4pq

)(3.4.2c)

Q4 : p(uu+ uu)− q(uu+ uu) =

=pQ− qP1− p2q2

((uu+ uu)− pq(1 + uuuu)

)(3.4.2d)

where P 2 = p4 − γp2 + 1 , Q2 = q4 − γq2 + 1 .

H-list:

H1 : (u − u)(u − u) = p2 − q2 (3.4.3a)

H2 : (u − u)(u − u) = (p− q)(u+ u+ u+ u) + p2 − q2 (3.4.3b)

H3 : p(uu+ uu)− q(uu+ uu) = δ2(p2 − q2) (3.4.3c)

A-list:

A1 : p (u+ u)(u + u)− q(u + u)(u + u) = δ2pq(p2 − q2) (3.4.4a)

A2 : p(1− q2)(uu+ uu)− q(1− p2)(uu+ uu) + (p2 − q2) (1 + uuuu) = 0

(3.4.4b)

The Q-equations are related through the following coalescence diagram, i.e. displayinghow the equations relate to eachother through certain limits on the parameters:

Q4

Q2 (Q1)δ=0

Q3

Q1

(Q3)δ=0

Soliton type solutions have been constructed by now for all equations in the ABS list [Nijhoff,Atkinson & Hietarinta, 2009; Atkinson & Nijhoff, 2010].


Remarks: It is in itself remarkable that the list of equations found is so short. In fact,all equations in the list Q are in fact special subcases of the last equation, Q4, which wasdiscovered by V. Adler in 1997. The latter equation can also be expressed as:

A [(u− b)(u− b)− (a− b)(c− b)][(u− b)(u− b)− (a− b)(c− b)

]+

+B [(u− a)(u − a)− (b − a)(c− a)][(u− a)(u − a)− (b − a)(c− a)

]=

= ABC(a− b) (3.4.5)

cf. [9]. Here the extra parameters c, C are related through C2 = 4c3 − g2c − g3 and to(a,A) and (b, B) through the relations:

A(c− b) +B(c− a) = C(a− b) ,

a+ b+ c =1

4

(A+B

a− b

)2

.

A third alternative form of Adler’s equation reads as follows:

sn(α)(vv + vv

)− sn(β)

(vv + vv

)− sn(α− β)

(vv + vv

)

+ksn(α)sn(β)sn(α− β)(1 + vvvv

)= 0 , (3.4.6)

[Hietarinta, 2005] which is somewhat simpler. In (3.4.6) the sn denote the Jacobi ellipticfunctions, cf . Appendix A, of modulus k, i.e. sn(α) = sn(α; k), etc.

It should be noted that all previous examples presented in section 3.2 can be recognisedas special subcases of the equations in the lists Q,H and A, and which can be obtained by“degeneration” of Adler’s equation in either of its forms. All these equations possess Laxpairs, but it was only recently that explicit solutions were found for most of the new casesincluding of Adler’s equation, cf. e.g. [12].

Adler’s equation is the discrete analogue of a famous soliton equation discovered in 1980by I. Krichever and S. Novikov, which reads:

ut = uxxx −3

2

u2xx − (4u3 − g2u− g3)ux

, (3.4.7)

and which generalises the Schwarzian KdV equation.

3.5 Lattice KdV Equation

We finish this Lecture by deriving the lattice analogue of the KdV equation, in contrast tothe lattice potential KdV equation (2.2.54) (or equivalently (3.2.4)):

(wn,m − wn+1,m+1)(wn,m+1 − wn+1,m) = p2 − q2 . (3.5.1)

We recall that the difference between the continuous KdV equation (2.2.30) and its potentialversion (2.2.39), as discussed in Lecture 2, was simply that its solutions were connected

3.6. SINGULARITY CONFINEMENT 63

through taking a derivative, u = wx. However, on the lattice there are many ways inwhich can do the analogue of “taking the derivative”, for instance we can replace it bytaking a difference involving two neighbouring vertices, like ∆nwn,m ≡ wn+1,m − wn,m or∆mwn,m ≡ wn,m+1 − wn,m , but these are not the only choices. Alternatively we can takea difference between vertices farther away or across diagonals, such as:

∆wn,m = wn+1,m+1 − wn,m or ∆wn,m = wn,m+1 − wn+1,m ,

or a host of other choices. As long as a well-chosen continuum limit reduces these to aderivative they can be justifiably be considered to be the discrete analogues of the operationof taking a partial derivative. The most sensible way to make a choice is to look at theequation at hand and then decide what would be for the given equation the most naturalchoice to apply in that case.

In the case of the lattive potential KdV equation (3.5.1) the natural choice of the discreteanalogue of the derivative seems to be either one of the choices given above, namely adifference across the diagonal. This suggest that as lattice KdV variables we would takeeither one of the two choices:

Q = w − w or R = w − w . (3.5.2)

It is then straightforward from (3.2.4) (using the notation in terms of ˜- and shifts) toderive the following equation for Q:

Q− Q =a

Q− a

Q

⇔ R− R =a

R− a

R, (3.5.3)

where a ≡ p2− q2. The equations for Q and R are simply related by the fact that QR = a.A Lax pair for either of the equations (3.5.3) can be easily derived, by starting from

the Lax pair for the potential KdV equation. In fact, rather than writing it as a first ordermatricial system, one may derive from the Lax pair a three-point linear equation in termsof one of the components of the vector φ on which the Lax matrices act. In this way weobtain the following scalar Lax pair:

ϕ = Q ϕ+ Λϕ , (3.5.4a)

ϕ = ϕ+Rϕ , (3.5.4b)

remembering that QR = a.

Exercise 3.5.1. Derive the Lax pair (3.5.4) from the set of equations (3.3.2) (with γ =γ′ = 1) by setting ϕ = G and identifying the spectral variable by Λ = k2 − q2.

Exercise 3.5.2. Show that the consistency condition of the Lax pair (3.5.4) viewed as anoverdetermined discrete linear system, leads to the lattice KdV equation (3.5.3).

3.6 Singularity confinement

In the seminal paper of B. Grammaticos et al. another point of view on integrability wasdeveloped than the one discussed in section 3.2, namely based on the issue of well-posedness


of initial value problems on the lattice. It is based on a phenomenon happening in integrablediscrete systems, which nowadays is called singularity confinement and which describes whathappens with singularities of the solutions of discrete system in the space of initial values.

Proposition: In an integrable lattice equation of KdV type singularities induced by initialdata do not propagate.

As an example let us consider the lattice KdV equation (3.5.3), (setting for conveniencethe parameter a = 1 w.l.o.g.)

Rn+1,m+1 −Rn,m =1

Rn+1,m− 1

Rn,m+1.

We will now study how the the initial data progresses, when one hits a singularity. Theinitial data is given at the solid line on Figure 3.1, a, b, 0, c, d. When one proceeds fromthese initial values one obtains infinity at two places, then one 0 at the next level and finallytwo ambiguities of the type ∞−∞.

A more detailed analysis with the initial value 01 = ε (small) yields the following valuesat the subsequent iterations

∞1 = b+1

ε− 1

a, ∞2 = c+

1

d− 1

ε,

at the first stage, and on the next

s = a+1

∞1− 1

f, t = d+

1

g− 1

∞2,

02 = ε+1

∞2− 1

∞1= −ε+

(b− c+ 1

a− 1

d

)ε2 + . . .

d

c

01

b

a

f g∞1 ∞2

s t02

?1 ?2

Figure 3.1: Propagation of singularities in a 2D map. Here n grows in the SE direction andm in the SW direction.

3.6. SINGULARITY CONFINEMENT 65

Then at the next step we can resolve the ambiguities:

?1 = ∞1 +1

02− 1

s= c+

1

d− 1

a− 1/f+ O(ε)

?2 = ∞2 +1

t− 1

02= b− 1

a+

1

d+ 1/g+ O(ε)

Thus the singularity is confined.

Note that if the lattice equation is deformed, e.g. by taking:

Rn+1,m+1 −Rn,m =1

Rn+1,m− λ

Rn,m+1

with λ 6= 1, the above fine cancellation would no longer happen, and singularities wouldagain occur at ?1 and ?2, and would persist throughout! In that case the singularities are nolonger confined to a finite number of iteration steps, and we conclude that the correspondingmap is not integrable.

Remark: Another point of view on 2D singularity confinement is provided by the require-ment of “ultra-local” singularity confinement (R. Sahadevan and H. Capel, 2003): One onlyassumes a singularity at one point and requires regularity at all other points.

Referring to Figure 3.2 we have initial values are at black disks, the values at opencircles are determined from them. The initial values can be chosen so that u11 =∞, and therequirement is that u12, u21 are finite and ambiguity at u22 can be resolved using ǫ-analysis.For example for the lattice potential KdV in the form

wn+1,m+1 = wn,m −1

wn,m+1 − wn+1,m,

the initial value w01 = w10 implies

w11 = ∞, , w12 = w01,

w21 = w10, , w22 =∞−∞ ?

w00

w01

w02

w20

w21

w22

w10

w11

w12

Figure 3.2: Setting for “ultra-local” singularity confinement.


and a detailed analysis reveals that if w10 = w01 + ǫ then

w11 = 1ǫ + w00,

w12 = w01 + ǫ+ ǫ2(w02 − w00) + O(ǫ3),

w21 = w01 + 0 + ǫ2(−w20 + w00) + O(ǫ3),

w22 = w11 −1

w12 − w21

=1

ǫ+ w00 −

1

ǫ+ ǫ2(w02 + w02 − 2w00) + O(ǫ3)

= w02 + w20 − w00 + O(ǫ)

This resolves the ∞−∞ singularity, and recovers initial value w00 which was temporarilysubmerged to order ǫ2.

Literature

1. M.J. Ablowitz and F.J. Ladik, A nonlinear difference scheme and inverse scattering, Stud.Appl. Math. 55 (1976) 213—229; On the solution of a class of nonlinear partial differenceequations, ibid. 57 (1977) 1–12.

2. R. Hirota, Nonlinear partial difference equations I-III, J. Phys. Soc. Japan 43 (1977), 1424–1433, 2074–2089.

3. E. Date, M. Jimbo and T. Miwa, Method for generating discrete soliton equations I-V, J.Phys. Soc. Japan 51 (1982) 4116–4131, 52 (1983) 388–393, 761–771.

4. B. Grammaticos, A. Ramani and V. Papageorgiou, Do integrable mappings have the Painleveproperty?, Phys. Rev. Lett. bf 67 (1991) 1825–1828.

5. F.W. Nijhoff, G.R.W. Quispel and H.W. Capel, Direct linearization of nonlinear difference-difference equations, Phys. Lett. 97A (1983) 125–128; G.R.W. Quispel, F.W. Nijhoff, H.W.Capel and J. van der Linden, Linear integral equations and nonliner difference-differenceequations, Physica 125A (1984) 344–380.

6. F.W. Nijhoff and H.W. Capel, The discrete Korteweg-de Vries equation, Acta ApplicandaeMathematicae 39 (1995) 133–158.

7. F.W. Nijhoff and A.J. Walker, The discrete and continuous Painleve VI hierarchy and theGarnier systems, Glasgow Math. J. 43A (2001) 109–123.

8. V.E. Adler, A.I. Bobenko and Yu.B. Suris, Classification of integrable equations on quad-graphs, Commun. Math. Phys. 233 (2003) 513–543.

9. F.W. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. 297A

(2002), 49–58.

10. J. Hietarinta, Searching for CAC-maps, J. Nonlin. Math. Phys. 12 Suppl. 2 (2005) 223–230.

11. F.W. Nijhoff, J. Atkinson and J. Hietarinta, Soliton Solutions for ABS Lattice Equations: ICauchy Matrix Approach, J. Phys. A: Math. Theor. 42 40 (2009) 404005 (34pp).

12. J. Atkinson and F.W. Nijhoff, A constructive approach to the soliton solutions of integrablelattice equations, Commun. Math. Phys. 299 (2010) 283–304.

Lecture 4

Special Solutions & ContinuumLimits

Integrable nonlinear equations stand out among other differential or difference equationsby the fact that one can construct infinite classes of solutions in explicit form, i.e. exactsolutions which can be explicitely written down. This distinguishes them from most othermodel equations that occur in the applied mathematical sciences.

The partial difference equations (P∆Es) that we have considered in Lecture 3 admitseveral types of special solutions. Such solutions include:

• rational solutions;

• soliton solutions;

• periodic solutions;

• similarity solutions.

These different solutions require different techniques to obtain them, and to write themdown in terms of explicit formulae. For example, to obtain soliton solutions we can usethe inverse scattering transform method, or alternatively the technique of Backlund trans-formations, whilst period solutions rely on a method called finite-gap integration, whichrelies on techniques from algebraic geometry. In most cases the problem of obtaining suchspecial solutions amounts to the investigation of reductions of the P∆Es: the choice of theclass of solutions restricts the parameter-space from an infinite-dimensional space to a finite-dimensional one. Effectively this means that the original P∆E is reduced to an ordinarydifference equation (O∆E) or a system of O∆Es in the process of obtaining the solution,which could often be expressed through the imposition of additional compatible constraints.The reduced equations, can then be studied in their own right, and they may take differentshapes, such as:

• discrete-time equations of motion (of a many-body system);

• dynamical mappings;

67

68 LECTURE 4. SPECIAL SOLUTIONS & CONTINUUM LIMITS

• ordinary difference equations of Painleve type,

which then can subsequently solved from that perspective. Thus, we get a solution schemethat looks like:

P∆E → O∆E system → explicit solution .

In this lecture we will concentrate on two types of special solutions: i) single soliton solutionsobtainable through BTs; ii) simple periodic solutions as dynamical mappings.

4.1 Solutions from discrete Backlund transforms

There are several ways to obtain soliton solutions: the inverse scattering method, the Hirota’sdirect method and the use of Backlund transformations, see e.g. the textbooks [Lamb, 1980;Ablowitz & Segur, 1982; Dodd, Eilbeck, Gibbon & Morris, 1982; Newell, 1983; Drazin &Johnson, 1989]. Here we will focuss primarily on the latter method, which in the discretecase is particularly natural, because in some sense the equations constituting the BTs are,in fact, copies of the lattice equations themselves! There are actually two types of BTs:

• auto-Backlund tranforms, i.e. nonlinear transformations that bring you from one so-lution, of the P∆E under consideration, to another solution of the same equation;

• non-auto-Backlund tranforms, which bring you from a given solution of a P∆E to asolution of a different P∆E. In fact, the Miura transformations we have encounteredpreviously, fall in this class.

In either case, one of the first problems we encounter is to find an initial solution, whichcould be a trivial solution (like the zero solution, if the equation admits it), but sometimesit might be problematic to find such a simple solution. Whenever we have such a solution,we can use it as a ”seed solution” on which we can implement the BT. Here we will givesome examples, both in the case of a non-auto-BT, as well as in the case of an auto-BT,how this works in the discrete case (the continuous case was already treated in Lecture 2).

4.1.1 Non-auto-BTs for P∆Es

We have encountered such non-auto-BTs as ”Miura transformations” before, between variousmembers of the KdV class of differential equations, as well as of the difference equations,e.g. the relations (3.2.6) between the lattice potential KdV and MKdV equations, andthe relations (3.2.8) between the lattice potential MKdV and the lattice Schwarzian KdVequation.

We will give another example of such a non-auto-BT between two P∆Es, and show howit can be used to generate a solution of one of the two equations provided one knows asolution of the other. The example we will use is between the equations H1 (which coincideswith our standard example (2.2.46)) and H2 of the ABS list presented in subsection 3.4, i.e.(3.4.3b). Consider the relations:

−2ww = p2 + v + v , (4.1.1a)

−2ww = q2 + v + v , (4.1.1b)

4.1. SOLUTIONS FROM DISCRETE BACKLUND TRANSFORMS 69

which was first given in [Atkinson, 2008]. By eliminating the variable w from both equations(4.1.1a) and (4.1.1b), using shifts in the variables n and m on the lattice, it is easy to showthat v obeys the lattice equation

(v − v)(v − v) = (p2 − q2)(v + v + v + v + p2 + q2

). (4.1.2)

Similarly, by eliminating v from the eqs. (4.1.1) it follows that w obeys the lattice potentialKdV equation

(w − w)(w − w) = (p2 − q2) . (4.1.3)

We can now take a known solution of (4.1.3), for instance the solution

w = wn,m = w0 + np+mq , w0 = constant .r.t. n and m , (4.1.4)

and insert this into the relations (4.1.1). Thus, making explicit the dependence on thevariables n and m, we obtain from (4.1.1a)

p2 + vn,m + vn+1,m = −2(w0 + np+mq)(w0 + (n+ 1)p+mq)

⇒ vn,m + vn+1,m = −(w0 + np+mq)2 − (w0 + (n+ 1)p+mq)2

where the latter equality follows from direct computation. The latter equation being anO∆E of the form yn + yn+1 = 0 , where yn = vn,m + (w0 + np +mq)2 , we can directlyintegrate to obtain a solution of the form yn = c(−1)n, with c a constant w.r.t. the variablen. Thus, we obtain

vn,m = cm(−1)n − (w0 + np+mq)2 ,

where cm is constant w.r.t. n (but not necessarily w.r.t. m!) as a solution of the first relationof the non-auto-BT (4.1.1). Subsequently, we have to satisfy also the second relation of thenon-auto-BT, but in this case, due to the symmetry of the relations as well as of the initialsolution, it is easy to see that the above expression for vn,m also yields a solution to thesecond relation, i.e. (4.1.1b), provided we take cm = c(−1)m with now c to be constant alsow.r.t. the variable m Hence, we arrive at the solution of H2, (4.1.2), namely

vn,m = c(−1)n+m − (w0 + np+mq)2 . (4.1.5)

Note that the solutions (4.1.4) and (4.1.5) can be extended to solutions on a multidimensionallattice by assuming that the initial value w0 depends on additional variables associated withadditional lattice directions. Hence, we can etend the solutions easily to a 3-dimensionallattice by setting

wn,m,h = u0 + np+mq + hr , vn,m,h = c(−1)n+m+h − (w0 + np+mq + hr)2 , (4.1.6)

and it is automatic that this solution will obey copies of the equations (4.1.3) and (4.1.2)on each 2-dimensional sublattice of the 3-dimensional lattice of variables n, m and h. Thus,we could call the solution (4.1.6) to be a covariantly extended solution of the quadrilaterallattice equation.


4.1.2 Backlund transformations for P∆Es

Let us consider the lattice equation

( w − w)(w − w) = p2 − q2 , (4.1.7)

for which we have seen in Lecture 3 that it possesses the C.A.C. property. This means thatwe can impose the same equation in any number of lattice directions of a multidmensionallattice, each direction of which carries its own lattice parameter. Thus, we can impose onthe same function w of independent discrete variables the additional equations;

(w − w)(w − w) = p2 − k2 , (4.1.8a)

(w − w)(w − w) = q2 − k2 . (4.1.8b)

and we interpret these equations as a BT u 7→ u of a given solution of (4.1.7) to a newsoution, in much the same way as eqs. (2.2.40) of Lecture 2 form the BT of the continuousequation, namely the ordinary potential KdV equation.

Exercise 4.1.1. By using the compatibility condition w = ˜w arising from shift on the twoequations (4.1.8), show that if w is a solution of (4.1.7) it follows that w is a solution of thesame equation.

In the same way that, once the inital solution w is given, the spatial part of the BT(2.2.41) is a Riccati equation (and hence linearisable), in the discrete case both parts ofthe discrete BT (4.1.8) are discrete Riccati equations. In a similar way as we have donein subsection 2.2.4 we shall now show how to obtain the 1-soliton solution of (4.1.7) byimplementation of the discrete BT.

Explicit example: one-soltion solution from discrete BT

The solutiom we start off with as a seed solution for the BT is the simple one we haveencountered before (see Lecture 3):

w(0)n,m = ξ0 + np+mq .

In connection with the C.A.C. property which establishes the multidimensional consistencyof the equation, the main thing to recognise now is that we need also multidimensionalconsistency of the solution. This means that additional variables associated with otherlattice directions, may be hidden in the initial value ξ0, in such a way that, for instance,ξ0 = ξ0+k . Taking this into account, we can now search for a form of the solution wn,m ofthe BT (4.1.8), which we assume to be in the form

wn,m = w(0)n,m +

fn,mgn,m

, (4.1.9)

where the factorisation into the functions f and g wll be taken such that we get a linear setof equations for these two functions (based on the fact that eqs. (4.1.8) is a set of discrete

4.1. SOLUTIONS FROM DISCRETE BACKLUND TRANSFORMS 71

Riccati equations). Substituting (4.1.9) into (4.1.8a) we get

(w

(0)+f

g− w(0)

) (w(0) − w(0) − f

g

)= p2 − k2

⇒ −(w

(0) − w(0)) fg+(w(0) − w(0)

) fg=f

g

f

g

⇒ f

g=

(w

(0) − w(0))f

−f +(w(0) − w(0)

)g,

and this can be split into the following (matricial) linear system

(fg

)=

(w

(0) − w(0) , 0

−1 , w(0) − w(0)

)(fg

)=

(p+ k 0−1 p− k

)(fg

),

and a simlar computation holds for the equation for the vector (f, g)T in the other latticedirection following from (4.1.8b). Thus, we obtain in both lattice directions the linearmatricial equations

φn+1,m = Pφn,m , φn,m+1 = Qφn,m , (4.1.10)

with

P =

(p+ k 0−1 p− k

), Q =

(q + k 0−1 q − k

).

Solving this system would lead to the solution of the BT, and this can be done on the basisof the following observations:

1. the two equations in (4.1.10) can be simultaneously solved, because in fact the two

matrices P and Q commute, PQ = QP (check this!), which implies thatφ =

˜φ.

Thus, the solution of the vector φ is simply given by φn,m = P nQmφ0,0 in terms ofan initial vector φ0,0.

2. crucially the matrices P and Q are lower triangular constant matrices, which impliesthat it is easy to compute arbitrary powers of such matrices, following the general rule:

A =

(a 0c b

)⇒ AN =

(aN , 0

c aN−bN

a−b , bN

).

(Prove this statement by induction!).

Evaluating the product P nQm we, thus, obtain the following result:

(fn,mgn,m

)=

((p+ k)n(q + k)m , 0

− 12k [(p+ k)n(q + k)m − (p− k)n(q − k)m] , (p− k)n(q − k)m

)(f0,0g0,0

).

(4.1.11)The Backlund transformed solution un,m can now be extracted from (4.1.11), simply readingoff the functions fn,m, gn,m and expressing their ratio in terms of the initial value ratio


f0.0/g0,0 and thus reconstituting (4.1.9). The end result, after some simple algebra, is the

solution w(1)n,m ≡ wn,m given by

wn,m = ξ0+k+np+mq+ρn,m(w0,0 − ξ0 − k)

1− 12k (w0,0 − ξ0 − k)(ρn,m − 1)

, ρn,m ≡(p+ k

p− k

)n (q + k

q − k

)m

,

(4.1.12)and this is the 1-soliton solution of the lattice equation (4.1.7). Note that we see the returnof the plane-wave factors ρn,m which is the ingredient in this exact solution through whichthe main dependence on the independent discrete variables n, m enters in the solution.

Exercise 4.1.2. Compare the 1-soliton solution (4.1.12) of the lattice potential KdV equa-tion with the 1-soliton solution (2.2.43) of the continuous equation, by recalling that thediscrete plane wave factors ρ become continuous exponential functions after a continuumlimit, (see section 4.4).

Finally, the solution of the actual lattice kdV equation, which according to section 3.5 isthe equation (3.5.3) in terms of either Qn,m = wn,m−wn+1,m+1 or Rn,m = wn,m+1−wn+1,m,cf. (3.5.2), can obviously be obtained directly from (4.1.12) in explicit terms.

Remark: Higher soliton solutions for the lattice equation (4.1.7) can be obtained by it-erating this procedure, or more simply, by applying the lattice equation once again, butnow viewed as a permutability condition from three known solutions, namely the seed solu-tion w(0), the 1-soliton solution w(1) with parameter k = k1 and another 1-soliton solutionw(2) with prameter k = k2, to yield a new 2-soliton solution w(12) (depending on the twoparameters k1 and k2) according to the diagram:

w(0)

w(1)

w(2)

k1

k2

w(12)

k2

k1

This procedure is, however, not the most effective one to obtain higher soliton solutions,and alternative direct methods can yield these solutions in closed form, using special typesof determinants. (See, e.g., the results in [12]).

4.2 Continuum limits

One interest in difference equations arise from the fact that they appear as finite-differenceapproximations to differential equations, e.g. in Numerical Analysis. In that case one wouldbe intereted to make sure that a well-defined continuum limit yields the continuous equationthat one is interested in studying. Even if discretization of given differential equations is notthe main motivation for investigating many of the equations we have presented so far, onemay be interested in the questions: what continuous equation corresponds to the discrete

4.2. CONTINUUM LIMITS 73

equation we are looking at? Thus, studying systematically continuum limits of given P∆Esis of interest anyway. In doing this there a two important features to take on board: i)Continuum limits are not unique – one can have more than one differential equation arisingfrom one and the same difference equation; ii) in performing a continuum limits one shouldrespect the solution structure – making sure that the continuum limit of the equationsreflects a limit on the solutions as well. To make sure this is the case, we will motivate thelimits by investigating first what happens to the corresponding linear equations associatedwitht he nonlinear equations of interest.

4.2.1 Plane-wave factors and linearisation

As an example for our ideas we will study the continuum limits of (2.2.46), i.e. the P∆E as-sociated with the Backlund transformations of the KdV equation, which we constructed inLecture 2. In Lecture 3 we pointed out that the Backlund parameters λ and µ, which can beassociated with the directions of the lattice, could be interpreted as parameters measuringsomehow the grid size in each direction. It is, thus, these parameters that can be used as“tuning parameters” by which the lattice can be shrunk or expanded in a certain direction,eventually allowing us to shrink the lattice points together to create a continuum of points.

However, there may be many different ways in which this can happen and, thus, inprinciple there may be various limits that we could perform on a given lattice equation. In acontinuum limit involving a lattice parameter (or step-size parameter), say h, the operationof a difference such as (0.0.4) will tend to a derivative, namely by (0.0.3), as was explained inLecture 1. Thus, it is by doing Taylor expansions on the shift operators like (0.0.5), namelyby inserting in the equations expansions of the form:

y(x+ h) = y(x) +h

1!

dy

dx+h2

2!

d2y

dx2+ · · · (4.2.1)

into the difference equations, and then by expanding power-by-power in the lattice param-eter h that we obtain the transition from difference to differential equation. Typically theequation that emerges as “the continuum limit” of the discrete equation is the coefficient ofthe dominant term in this expansion as h→ 0.

Performing this sequence of steps on a given lattice equation there are two questions toanswer, namely

• among the various parameters present in the lattice equation, how do we identify theparameter (or combination of parameters) to take as the one tending to zero in orderto shrink the lattice?

• how do we determine the behaviour of the independent and dependent variables underthe limit on this chosen parameter?

In general a brute force or naive continuum limit may easily lead to a total collapse of theequation, where the Taylor expansions applied to the equation lead to mismatch of orders,and hence to a situation where the limit results in no equation at all (due to conflictingconstraints emerging from the expansion) or to a trivial equation in leading order in thelattice parameter. To avoid this problem and to answer the questions above, it turns out itis useful to first study the continuum limits of the linearised equation before we attack the


full nonlinear equation. We will, thus, first derive the linearised form of the lattice equationsunder consideration and derive a special class of solutions of these linear equations to usethese as a guidance on how to take nontrivial and consistent limits.

By a linearisation of a nonlinear lattice equation such as (2.2.46) we mean the linearequation obtained by expanding the dependent variable around a specific known solutionof the nonlinear equation and taking the dominant term in the expansion. The simplestlinearisations are obtained by taking a trivial solution, such as the zero solution (if it exists).In the case of the example (2.2.46), taking w ≡ 0 is not allowed since it does not lead to asolution of that equation, but we can modify the equation slightly by changing it into

(p− q + un,m+1 − un+1,m)(p+ q + un,m − un+1,m+1) = p2 − q2 , (4.2.2)

which is (3.2.4) in explicit form, by setting as before 4λ = p2, 4µ = q2, and wn,m =un,m − np−mq . It is easy to see that (4.2.2) admits the solution un,m ≡ 0, i.e. u vanishesfor all n,m. By setting next

un,m = ǫρn,m ,

and expanding up to linear terms in the small parameter ǫ, we obtain the following linearequation for ρ:

(p+ q)(ρn,m+1 − ρn+1,m) = (p− q)(ρn+1,m+1 − ρn,m) . (4.2.3)

It is easily verified that the linear lattice equation (4.2.3) obeys the consistency-around-the-cube property of section 3.2 in the same way as the full nonlinear equation, and hence we canconsistently embed this equation in a higher dimensional lattice by writing the compatiblesystem:

(p+ q)(ρ− ρ) = (p− q)(ρ− ρ) , (4.2.4a)

(p+ k)(ρ− ρ) = (p− k)(ρ− ρ) , (4.2.4b)

(q + k)(ρ− ρ) = (q − k)(ρ− ρ) , (4.2.4c)

where ¯ denotes the shift in the third direction associated with lattice parameter k. Thelinear system (4.2.4) has many solutions, but we will fix a specific class of solution bydemanding that the specific variable k is associated with a shift ρ 7→ ρ such that ρ = 0.(One way of thinking of what this means is that the solution ρ with this constraint is

obtained from an (inverse) Backlund transformation ρk7→ ρ with seed solution ρ = 0.)

Solving the two relations (4.2.4b) and (4.2.4c) with ρ = 0:

ρ =p− kp+ k

ρ , ρ =q − kq + k

ρ ,

we obtain the solution:

ρn,m =

(p− kp+ k

)n(q − kq + k

)m

ρ0,0 , (4.2.5)

with ρ0,0 some arbitrary initial value. It is straightforward to show by direct computationthat (4.2.5) is a solution of (4.2.3).

The solutions (4.2.5) we will refer to as lattice plane-wave factors and have already seenhow they appeared in section 4.1. Thus, not only do they play a role as approximated


solutions (namely as solutions of the linearised version of the nonlinear lattice equation(4.2.2)) but they also are the key ingredient in the exact solution of the full nonlinearequation. It is these solutions that we will exploit in the next section to formulate theprecise limits on parameters and discrete variables to get nontrivial limiting equations.

4.2.2 The semi-continuous limits

We will now study various limits we can perform on the plane wave factors ρ of the specificform found in (4.2.5). The guiding principle would be to seek ways in which this solutionwill approach exponential factors with continuous variables in the exponents. The “trick”to be used is the following limit which is well-known from basic analysis

limn→∞

(1 +

α

n

)n= eα .

Straight continuum limit

Focusing on one of the factors in (4.2.5) we can easily see that in the limit

m →∞ , q →∞ such thatm

qfinite (4.2.6)

we obtain:

limm→∞

q→∞

m=ξq

(q − kq + k

)m

= limm→∞

q→∞

m=ξq

(1 +

−2kq + k

)m

= limm→∞

(1 +

ξ

m

(−2k)(1 − kξ/m)

)m

= e−2kξ ,

(4.2.7)using the fact the extra term in the denominator within the brackets becomes negligeableas m→∞.

Let us now investigate the effect of this limit on the equations, e.g. the linear equation(4.2.4a). The idea is to re-interpret the dependent variable as

ρ = ρn,m =: ρn(ξ) , ξ = ξ0 +m

q, (4.2.8)

where ξ0 is some initial value. This means that any shift in the discrete variablem incrementsin the argument of the function ρ by 1/q. The next thing is for 1/q small, to consider Taylorseries expansions of the form:

ρn,m+1 = ρn(ξ +1

q) = ρn(ξ) +

1

q∂ξρn(ξ) +

1

2

1

q2∂2ξρn(ξ) + · · · ,

and inserting this into the eq. (4.2.4a) we get:

(1 +

p

q

)[ρn − ρn+1 +

1

q∂ξρn +

1

2

1

q2∂2ρn + · · ·

]

=

(−1 + p

q

)[ρn+1 − ρn +

1

q∂ξρn+1 +

1

2

1

q2∂2ρn+1 + · · ·

],


leading in dominant order (in terms of 1q ) to the differential-difference equation (D∆E):

∂ξ (ρn+1 + ρn) = 2p(ρn+1 − ρn) . (4.2.9)

This linear equation is a mixed form of differential equation (w.r.t variable ξ) and discrete(w.r.t. variable n, and is hence a semi-discrete equation. By construction, it can be directlyverified that the form:

ρn(ξ) =

(p− kp+ k

)n

e−2kξρ0(0) ,

is a solution of this equation.Inspired by this result, let us now turn to the nonlinear equation for u, (4.2.2), and

perform exactly the same limit there. Thus, introducing in a similar was as for the linearequation the reinterpretation of the discrete variables

un,m =: un(ξ) , ξ = ξ0 +m

q, (4.2.10)

and we derive by similar Taylor expansions as before:

p2 − q2 =

[p− q +

(un +

1

q∂ξun +

1

2

1

q2∂2un + · · ·

)− un+1

]×

×[p+ q + un −

(un+1 +

1

q∂ξun+1 +

1

2

1

q2∂2un+1 + · · ·

)],

Expanding in powers of 1/q, noting that the dominant terms of order O(q2) and order O(q)cancel identically, we obtain as coefficient of the leading term of order O(1) the followingequation

∂ξ (un + un+1) = 2p(un+1 − un)− (un+1 − un)2 , (4.2.11)

which is consequently the continuum limit of the lattice equation (4.2.2) under this limit asq →∞. Eq. (4.2.11) is a nonlinear D∆E, which is of first order in the derivative w.r.t. thecontinuous variable ξ and of first order in the discrete variable n as well.

Remark: By comparing eq. (4.2.11) with eq. (2.2.41) we recognise that by the change ofvariables x = 2ξ , p2 = 4λ2 , w−w = un+1−un−p , ∂ξun = 2wx , we recover the spatialpart of the BT of the KdV equation.

Other lattice equations: We can perform simlar limits on the other members of theKdV family of lattice equations, namely on (3.2.5) amd (3.2.7), leading to

∂ξ(vn+1vn) = p(v2n+1 − v2n) (4.2.12)

and

(∂ξzn)(∂ξzn+1) = p2(zn − zn+1)2 , (4.2.13)

respectively. We shall show in section 4.3 that all these D∆Es are integrable by virtue ofthe existence of semi-discrete analogues of the Lax pairs in all cases.


Skew continuum limit

The limit described above is not the only continum limit that we can perform on the latticeequation. Instead of taking a limit on one of the variables n and m separately, one canalso first mix them up, by means of a change of independent variables on the lattice, beforetaking a limit. As we shall see this will lead to quite different semi-continuous equation asa result of the limit.

To describe this, let us first consider the linearised equation (4.2.3) and the followingchange of variables:

ρn,m = Rn+m,m ⇒ ρn+1,m = RN+1,m , ρn,m+1 = RN+1,m+1 , ρn+1,m+1 = RN+2,m ,(4.2.14)

where N = n+m which can be visualised in the diagram:

R = ρ R = ρ

R = ρ

R = ρR

This change of independent variables (n,m) 7→ (N = n+m,m) bring the linear equation(4.2.4a) in the form

(p+ q)(RN+1,m+1 −RN+1,m) = (p− q)(RN+2,m+1 −RN,m) . (4.2.15)

Let us investigate what happens on the level of the plane-wave factors ρ given by (4.2.5)with this change of variables (n,m) 7→ (N = n+m,m) . Rearranging factors we get:

ρn,m =

(p− kp+ k

)n+m(q − kq + k

p+ k

p− k

)m

ρ0,0 =

(p− kp+ k

)N (1 +

2(q − p)k(q + k)(p− k)

)m

ρ0,0 := RN,m .

Keeping N fixed and setting δ = q − p , we can now perform the limit

n → −∞ , m→∞ , δ → 0 such that N fixed , δm finite

and focusing on what happens with the second factor in this limit we observe:

limm→∞

δ→0δm=τ

(1 +

2δk

(p+ δ + k)(p− k)

)m

= limm→∞

δ→0δm=τ

(1 +

2δk

(p2 − k2) + (p− k)δ

)m

= limm→∞

(1 +

τ

m

2k

(p2 − k2) + (p− k)τ/m)

)m

= exp

(2kτ

p2 − k2), (4.2.16)


using the fact the extra term in the denominator within the brackets becomes negligeable asm→∞. Thus, the limit (4.2.16) makes good sense on the level of the plane-wave factors.

To investigate what happens with the linear lattice equation (4.2.15) under the limit(4.2.16), we set

R = RN,m =: RN (τ) , τ = τ0 +mδ , (4.2.17)

(allowing for some constan background value τ0 of the continuous variable), and applyingthe Taylor expansion:

RN,m+1 = RN (τ + δ) = RN (τ) + δ ∂τRN (τ) +1

2δ2 ∂2τRN (τ) + · · · .

Inserting this into the eq. (4.2.15) we get:

(2p+ δ)

[(RN+1 + δRN+1 +

1

2δ2RN+1 + · · ·

)−RN+1

]

= −δ[(RN+2 + δRN+2 +

1

2δ2RN+2 + · · ·

)−RN

],

which in leading order O(δ) yields as dominant term in the expansion the equation:

2pRN = RN−1 −RN+1 . (4.2.18)

It is straightforward to check that

RN (τ) =

(p− kp+ k

)N

exp

(2kτ

p2 − k2)R0(0) , (4.2.19)

provides a solution of eq. (4.2.18).Let us now move to the nonlinear equation and perform a similar limit there. In fact,

applying the change of variables (n,m) 7→ (N = n+m,m) in (4.2.2) with the changes:

un,m = Un+m,m ⇒ un+1,m = UN+1,m , un,m+1 = UN+1,m+1 , un+1,m+1 = UN+2,m ,(4.2.20)

we obtain

(p− q + UN+1,m+1 − UN+1,m)(p+ q + UN,m − UN+2,m) = p2 − q2 , (4.2.21)

and then reinterpreting the variable U as

U = UN,m =: UN (τ) , τ = τ0 +mδ , (4.2.22)

we can perform a similar Taylor expansion on U as we did for R. Inserting this into (4.2.21)we get

[−δ +

(UN+1 + δUN+1 +

1

2δ2UN+1 + · · ·

)− UN+1

]×

×[2p+ δ + UN −

(UN+2 + δUN+2 +

1

2δ2UN+2 + · · ·

)]= −(2p+ δ)δ ,


which in leading order yields

(UN+1 − 1)(2p+ UN − UN+2) = −2p ,

or equivalently

UN = 1− 2p

2p+ UN−1 − UN+1. (4.2.23)

In spite of the fact that eq. (4.2.23) was derived starting from the same lattice equation,it is quite a different nonlinear D∆E from (4.2.11), as is evident by inspecting the orders:this equation is first order in the derivative w.r.t. the continuous variable τ , and secondorder w.r.t. the discrete variable N , whereas (4.2.11) is first order in both the discrete andcontinuous variables but the derivative w.r.t. ξ acts on both un and un+1. We will assess insection 4.3 that both equations (4.2.11) and (4.2.23) are integrable in the sense that thereexists an associated Lax pair in both cases.

Other examples: The process described above to obtain a mixed continuum limit werefer to as the skew continuum limit of the lattice equation. It can be applied also to theother members of the lattice KdV family, namely to eqs. (3.2.5) and (3.2.7). In those caseswe obtain

p∂τ logVN =VN−1 − VN+1

VN−1 + VN+1, (4.2.24)

and

ZN =2

p

(ZN−1 − ZN)(ZN − ZN+1)

ZN−1 − ZN+1(4.2.25)

respectively.

4.2.3 Full continuum limit

Finding the full continuum limit of lattice equations such as eq. (4.2.2) is a two-stage process:first we must establish a semi-discrete (semi-continuous) limit, as we have established in theprevious section, and second to find the full continuum limit from the latter in the next step.In this section we will perform the second step of the process starting from the skew limitof the equation, and do a second limit in order to turn the remaining discrete variable intoa continuous one. In fact, in order to obtain a nontrivial limit, we shall see that the processis slightly more involved , and that we need to mix once again both spatial and temporalvariables in order to obtain a nonlinear PDE, retaining the integrability.

As before we shall use the plane-wave factors to guide us in finding the limit we needto impose in order to get nontrivial equations from the semi-discrete one. Taking the skewlimit of the plane-wave factor, i.e. the form of the variable RN (τ) as given in (4.2.19), we


have

(p− kp+ k

)N

exp

(2kτ

p2 − k2)

= exp

{2kτ

p2 − k2 +N

[ln

(1− k

p

)− ln

(1 +

k

p

)]}

= exp

{2kτ

p2

[1 +

k2

p2+k4

p4+ · · ·

]− 2N

[k

p+

1

3

k3

p3+

1

5

k5

p5+ · · ·

]}

= exp

{2k

(τ

p2− N

p

)+ 2k3

(τ

p4− 1

3

N

p3

)+ 2k5

(τ

p6− 1

5

N

p5

)+ · · ·

}

−→ ekx+k3t+k5t′+··· .

This leads to the identification of the variables of the full continuum limit as the coefficientsof the various powers of k in the expansion of the exponents of the plane-wave factors,namely

x = 2

(τ

p2− ξ)

, t = 2

(τ

p4− 1

3

ξ

p2

), (4.2.26)

in which as before ξ = N/p . Thus, this calculation suggests, first, that the solution (4.2.19)can be identified as follows:

RN (τ) := R(ξ, τ) ⇒ RN±1(τ) = R(ξ ± 1

p, τ) = R± 1

pRξ +

1

2p2Rξξ ±

1

6p3Rξξξ + · · · ,

and, second, to perform the change of variables (4.2.26), i.e.

R(ξ, τ) := R(x, t) ,

which implies by chain rule:

∂R

∂ξ=∂R

∂x

∂x

∂ξ+∂R

∂t

∂t

∂ξ= −2Rx −

2

3p2Rt , (4.2.27a)

∂R

∂τ=∂R

∂x

∂x

∂τ+∂R

∂t

∂t

∂τ=

2

p2Rx +

2

p4Rt . (4.2.27b)

Since the two seps (i.e. Taylor expansion of the shift in N and change of variables (ξ, τ) 7→(x, t)) involve the parameter p, the expansion in powers of 1/p and the selection of thedominant term can only be done after having done both of these steps. Thus, starting fromthe semi-discrete equation (4.2.18) we get

2p∂τR = −2

p∂ξR−

1

3p3∂3ξR − · · · ⇒

⇒ 2p

(2

p2∂x +

2

p4∂t

)R = −2

p

(−2∂x −

2

3p2∂t

)R− 1

3p3

(−2∂x −

2

3p2∂t

)3

R− · · · ,

and we observe that while the leading term of order O(1/p) cancels identically, the nextterm of order O(1/p3) yields precisely the equation:

Rt = Rxxx ,


as expected.Turning now to the nonlinear equation (4.2.23) and performing the same continuum limit

there, setting first

UN (τ) := U(ξ, τ) ⇒ UN±1(τ) = U(ξ ± 1

p, τ) = U ± 1

pU ξ +

1

2p2U ξξ ±

1

6p3Uξξξ + · · · ,

which by insertion in (4.2.23) yields:

∂τU = 1−[1− 1

2p

(2

pU ξ +

1

3p3U ξξξ + · · ·

)]−1

,

and, second, to use similar expressions as (4.2.27) to implement the change of variables(ξ, τ) 7→ (x, t) to obtain:

(2

p2∂x +

2

p4∂t

)U = 1−

[1− 1

p2

(−2∂x −

2

3p2∂t

)U − 1

6p4

(−2∂x −

2

3p2∂t

)3

U − · · ·]−1

= −[1

p2

(−2∂x −

2

3p2∂t

)U +

1

6p4

(−2∂x −

2

3p2∂t

)3

U + · · ·]−[1

p2

(−2∂x −

2

3p2∂t

)U + · · ·

]2− · · ·

expanding the demoninator on the right-hand side in a power series. Again we observethat the terms of order O(1/p) cancel identically, and that the next dominant orderO(1/p3) yields the equation:

Ut = Uxxx − 3U2x , (4.2.28)

which is the potential KdV equation (coinciding with (2.2.39) in terms of the variable w upto a change of sign in the independent variables, i.e. after x 7→ −x, t 7→ −t we recover theequation for w).

We see that we have come now full circle: we started out with the continuous KdVequation, derived its Backlund transformations, which using the permutability theorem ledto the construction of a lattice of solutions, the relations between these solutions beingreinterpreted as a partial difference equation on the twodimensional lattice. As a dynamicalequation the latter was seen as a discretisation of some continuum equations, both semi-discrete as well as fully discrete, and the full continuum limit now turns out to be theequation we started out from, namely the KdV itself. Perhaps this is not so surprising,but what is remarkable is that the equations at all levels are compatible with eachother:the continuous and discrete equations can be imposed simultaneously on one and the samevariable u, and, in fact, the variable U of (2.2.39) is nothing else than the variable w ofLecture # 2.

Remark: Performing similar continuum limits on the semi-discrete equations (4.2.24) and(4.2.25) we recover the following fully continuous equations, namely

Vt = Vxxx − 3VxVxxV

, (4.2.29)


which is the potential MKdV equation, and

Zt = Zxxx −3

2

Z2xx

Zx, (4.2.30)

which is the Schwarzian KdV equation.

Literature

1. G.L. Lamb, Elements of Soliton Theory, (Wiley Interscience, 1980).

2. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM, 1982).

3. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equa-tions, (Academic Press, 1982).

4. A.C. Newell, Solitons in Mathematics and Physics, (SIAM, 1983).

5. P.G. Drazin and R.S. Johnson, Solitons. An Introduction, (Cambridge Univ. Press, 1989).

6. T. Miwa, M. Jimbo and E. Date, Solitons, Differential Equations, Symmetries and InfiniteDimensional Lie Algebras, (Cambridge University Press, 2000).

7. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scat-tering, LMS Series (Cambridge Univ. Press, 1991).

8. R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,Phys. Rev. Lett. 27 (1971), 1192.

9. R. Hirota, Direct Method for finding exact solutins of nonlinear evolution equations, Lect.Notes in Math. 515 (Springer Verlag, 1976), p. 40.

10. R. Hirota, Direct Methods in Soliton Theory, in: R.K. Bullough and P.J. Caudrey, eds.,Solitons, (Springer Verlag, 1980).

11. J. Atkinson, Backlund transformations for integrable lattice equations, J. Phys. A: Math.Theor. 41 (2008) 135202 (8 pp.).

12. J. Atkinson, J. Hietarinta and F.W. Nijhoff, Seed and soliton solutions for Adler’s latticeequation, J. Phys. A: Math. Theor. 40 1 (2007) F1–F8.

13. J. Atkinson, J. Hietarinta and F.W. Nijhoff, Soliton solutions for Q3, J. Phys. A: MathTheor. 41 14 (2008) 142001 (11pp)

14. V.G. Papageorgiou, F.W. Nijhoff and H.W. Capel, Integrable Mappings and Nonlinear Inte-grable Lattice Equations, Physics Letters 147A (1990), 106–114.

15. H.W. Capel, F.W. Nijhoff, H.W. Capel, Complete Integrability of Lagrangian Mappings andLattices of KDV Type, Physics Letters 155A (1991), 377–387.

16. G.R.W. Quispel, H.W. Capel, V.G. Papageorgiou and F.W. Nijhoff, Integrable Mappingsderived from Soliton Equations, Physica 173A (1991), 243–266.

Contents

1 Elementary Theory of Difference Equations 11

1.1 Elementary Difference Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Difference versus differential operators . . . . . . . . . . . . . . . . . . 11

1.1.2 The finite-difference operator . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.3 Difference analogues for some familiar functions . . . . . . . . . . . . . 14

1.1.4 Discrete integration: the antidifference operator . . . . . . . . . . . . . 16

1.1.5 ∗ q-Difference operators and Jackson integrals . . . . . . . . . . . . . . 17

1.2 Linear (finite-)difference equations . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Linear constant coefficient difference equations . . . . . . . . . . . . . 18

1.2.2 Linear first order ∆Es with nonconstant coefficients . . . . . . . . . . 21

1.2.3 Higher order linear difference equations and linear (in)dependence . . 22

1.2.4 Further techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 A simple nonlinear difference equation – discrete Riccati equation . . . . . . . 27

1.3.1 Continuous Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.2 Discrete Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Partial Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4.1 Discretization and boundary values . . . . . . . . . . . . . . . . . . . . 31

1.4.2 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 From Continuous to Discrete Equations via Transformations 35

2.1 Special Functions and linear equations . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 Weber Functions and Hermite Polynomials . . . . . . . . . . . . . . . 36

2.1.2 Darboux and Backlund transformation in general . . . . . . . . . . . . 38

2.1.3 Another example: Bessel Functions . . . . . . . . . . . . . . . . . . . . 40

2.2 Backlund transformations for non-linear PDEs . . . . . . . . . . . . . . . . . 42

2.2.1 Lax pair for KdV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2 Miura transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.3 The Backlund transformation . . . . . . . . . . . . . . . . . . . . . . 44

2.2.4 Using BTs to generate multisoliton solutions . . . . . . . . . . . . . . 45

2.2.5 Permutability property of BTs . . . . . . . . . . . . . . . . . . . . . . 46

2.2.6 Backlund transformation for the sine-Gordon equation . . . . . . . . . 48

2.2.7 Transition to lattice equations . . . . . . . . . . . . . . . . . . . . . . 49

83

84 CONTENTS

3 Integrability of P∆Es 513.1 Quadrilateral P∆Es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Integrability and Consistency-around-the-Cube . . . . . . . . . . . . . . . . . 543.3 Lax pair for lattice potential KdV . . . . . . . . . . . . . . . . . . . . . . . . 573.4 ∗ Classification of Quadrilateral P∆Es . . . . . . . . . . . . . . . . . . . . . . 603.5 Lattice KdV Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.6 Singularity confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Special Solutions & Continuum Limits 674.1 Solutions from discrete Backlund transforms . . . . . . . . . . . . . . . . . . . 68

4.1.1 Non-auto-BTs for P∆Es . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Backlund transformations for P∆Es . . . . . . . . . . . . . . . . . . . 70

4.2 Continuum limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.1 Plane-wave factors and linearisation . . . . . . . . . . . . . . . . . . . 734.2.2 The semi-continuous limits . . . . . . . . . . . . . . . . . . . . . . . . 754.2.3 Full continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

math3491/5492: discrete systems and integrabilityfrank/math5492/lectures.pdf · 2010-12-08 ·...

Documents