gem software for computation of symmetries and
TRANSCRIPT
GeM software for computation of symmetries andconservation laws of differential equations
Alexei F. Cheviakov,Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2 Canada
E-mail: [email protected]
February 9, 2006
Abstract
We present a recently developed ”GeM” software package for automated symmetryand conservation law analysis of ordinary and partial differential equations. The pack-age contains a collection of easy-to-use yet powerful routines for mathematicians andapplied researchers. Determining systems are automatically generated and reduced,and in most cases completely solved. Classification of symmetries and conservationlaws without human intervention is implemented. The ”GeM” package is being success-fully used in ongoing research for analysis of previously intractable systems. Examplesare given.
Keywords: Symmetry; Conservation law; Symbolic computations; Classification.
1 Introduction.
Symmetry, adjoint symmetry and conservation law analysis are practically important meansof analysis of nonlinear ordinary and partial differential equations (DEs) and systems ofnonlinear DEs.
A symmetry of a system of DEs is any transformation of its solution manifold into itself. Asymmetry thus transforms any solution to another solution of the same system. Symmetriesof nonlinear partial differential equations can be used to reduce the order and/or number ofvariables, to obtain invariants and possibly linearize the problem (e.g. [1,2]), and to constructfamilies of new solutions from known ones (e.g. [?,4,5]). Self-similar solutions obtained fromsymmetries often have transparent physical meaning. Many appropriate examples can befound in [6].
One-parameter and multi-parameter Lie groups of point symmetries are found using thegeneral algorithmic Lie method of group analysis. It is equally applicable to algebraic andordinary and partial differential equations. In a similar algorithmic manner, contact andhigher-order symmetries are found (e.g. [2]). Closely related is the potential symmetrymethod [1] for discovering nonlocal symmetries of differential equations.
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It is well-known that Lie group analysis method is capable of detecting not only several-parameter, but also infinite-dimensional symmetry groups (depending on an arbitrary func-tions, see e.g. [7].)
For any system of DEs, from its admitted symmetry group, one can determine whether ornot it can be mapped into a linear system and find the explicit mapping [1, 8–10].
Though the Lie procedure is straightforward and applicable to any ODE/PDE system withsufficiently smooth coefficients, it often requires extensive algebraic manipulation and thesolution of (often large) overdetermined systems of linear PDEs. For many contemporarymodels, especially those that do possess non-trivial symmetry structure, such analysis presentsa significant computational challenge. The overview of the algorithmic procedure for localsymmetry group analysis and related challenges is presented in Section 2.1.
An important counterpart to knowledge of the symmetry structure of a DE system is infor-mation about its conservation laws. Conservation laws provide information about essentialphysical properties of the modelled process, and are essential for the PDE system analysis, inparticular, existence, uniqueness and stability analysis (e.g. [11–13]) and study of analytical(including geometrical) properties of solutions of PDE systems. The conservation law struc-ture also contains information about the possibility of linearization of the given DE systemby an invertible mapping and allows to construct the explicit mapping [14,15].
Several algorithms for finding conservation laws of PDEs and PDE systems exist [16]. Suchalgorithms often involve finding a vector of multipliers (also known as factors or characteristicfunctions) with prescribed dependence, such that a weighted linear combination of equationsof the system is a divergence expression. Fluxes may be found, for instance, from determiningequations [16] or by using integral formulas arising from homotopy operators [17–19].
Sets of symmetries and conservation laws for a given PDE system are closely related (see[20].) In particular, if a PDE system is self-adjoint (i.e. it can be obtained from a variationalprinciple), the multipliers of conservation laws are components a of symmetry generator.The action of a symmetry on a conservation law yields other (possibly new) conservationlaws [21].
Conservation laws can be used to introduce potential variables and thus yield nonlocally-related systems, which in turn can yield new nonlocal symmetries and conservation laws [22].
As for the Lie’s algorithm for finding local symmetries, conservation law construction meth-ods often involve the generation, simplification and solution of large overdetermined PDEsystems. For systems that contain arbitrary (constitutive) functions, one is interested in thequestion of conservation law classification with respect to specific forms of such functions,which presents even a greater computational challenge. For many PDE systems of interest,such analysis requires the use of symbolic software.
Modern analytical computation methods, based on Grobner bases [23,24] and characteristicsets [25,26], efficiently reduce large overdetermined systems of partial differential equations.For a review, see [27]. Many of these methods have been implemented on modern symbolicsoftware and are publicly available. In particular, the Rif package is now a part of WaterlooMaple.
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An algorithmic method of reduction of differential polynomial systems was suggested in [28].Its current implementation in Mathematica software requires regular user intervention andspecial knowledge. The method is being successfully used in current research for symmetryand conservation law computations and classification (e.g. [21].)
A general and one of the most powerful software packages for symmetry- and conserva-tion law-related computations is a REDUCE-based set of programs by Wolf [16]. It includesCRACK, LIEPDE and CONLAWi routines for PDE system reduction, search for symmetries,conservation laws and adjoint symmetries of DE systems.
A general-purpose Vessiot package for Maple for computations on jet spaces is availablein [29].
In this work, we introduce a recently developed package GeM [30]. The package containspractically useful routines to perform local (Lie, contact and higher-order) symmetry analysisand search for conservation laws of ordinary and partial differential equations without humanintervention (Section 3.) The use of package routines does not require special knowledge ofthe theory or continuous input. The package may be used in modern Maple versions (9.0and above) and, as Maple, is available for most popular platforms (UNIX, Linux, Windows.)
Package routines automatically generate systems of determining equations for conservationlaw/symmetry/adjoint symmetry analysis, which are then reduced and in most cases com-pletely solved. The routines employ a special symbolic representation of the system underconsideration, which leads to the overall significant speedup of computations. This factoris of particular importance in the problems of symmetry/conservation law classification forsystems with arbitrary constitutive function(s). The module routines can handle DE systemsof high order and with many equations and many dependent and/or independent variables.
Programs that use GeM routines, even ones involving extended classification, may be effec-tively run on a desktop computer.
Commands and features of the GeM package are discussed in Section 4. In Section 5, wepresent several run examples, including finding Lie and higher-order symmetries, conser-vation laws, and symmetry/conservation law classification for problems with constitutivefunctions.
2 Algorithms for obtaining local symmetries and con-
servation laws of systems of differential equations
2.1 The Lie procedure for finding symmetries.
For a general system of l partial differential equations of order p,
G(x, u, u1, ..., u
p) = 0,
G = (G1, . . . , Gl), x = (x1, . . . , xn) ∈ X , u = (u1, . . . , um) ∈ U ,
uk
=
(∂kuj
∂xi1 ...∂xik| j = 1, . . . , m
)∈ Uk, k = 1, . . . , p,
(2.1)
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the set of all solutions is a manifold Ω in (m+n) - dimensional space X×U , which correspondsto a manifold Ωp in the prolonged (jet) space X × U × U1 × · · · × Up of dependent andindependent variables together with partial derivatives [2].
Three types of local symmetries of DE systems are distinguished: point symmetries, contactsymmetries and higher-order symmetries.
1. Lie Point symmetries. The Lie method yields Lie groups of point transformations(symmetries)
(x′)i = f i(x, u, ε), i = 1, . . . , n,(u′)j = gj(x, u, ε), j = 1, . . . , m,
(2.2)
or, in the local form,
(x′)i = xi + εξi(x, u) + O(ε2), i = 1, . . . , n,(u′)j = uj + εηj(x, u) + O(ε2), j = 1, . . . , m,
(2.3)
that map solutions of (2.1) into solutions. Here ε is a group parameter, or a vector of groupparameters (in the case of multi-parameter groups.)
One considers the Lie algebra of infinitesimal operators (or symmetry generators)
X = ξi(x, u)∂
∂xi+ ηk(x, u)
∂
∂uk. (2.4)
(Here and below, summation with respect to repeating indices is assumed.) Each symmetrygenerator defines a vector field tangent to the solution manifold Ω of the system (2.1). Toeach symmetry generator (2.4) there uniquely corresponds a prolonged vector field
v = ξi ∂
∂xi+ ηk ∂
∂uk+ η
(1) ki
∂
∂uki
+ ... + η(p) ki1,...,ip
∂
∂uki1,...,ip
. (2.5)
tangent to the solution manifold Ωp in the jet space. Here η(w) ki1,...,ip
are the coordinates of
the prolonged tangent vector field corresponding to the derivatives of order w: uki1,...,iw ; w =
1, ..., p. The prolonged coordinates η(w) ki1,...,ip
are fully determined from (ξi, ηk) by differentialrelations
η(w) ki1,...,iw
= Diwη(w−1) ki1,...,iw−1
− (Diwξl)uki1,...,iw−1,l. (2.6)
Here it is assumed that η(0) k ≡ ηk. Di =∂
∂xi+ uk
i
∂
∂uk+ uk
ij
∂
∂uki
+ ... is a full derivative with
respect to xi.)
Once a symmetry generator (2.4) is known, the global Lie transformation group (2.2) isreconstructed explicitly by solving the initial value problem
∂f i
∂ε= ξi(f, g),
∂gk
∂ε= ηk(f, g), f i|ε=0 = xi, gk|ε=0 = uk. (2.7)
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To find all Lie group generators (2.4) admissible by the original system (2.1), one needs tosolve linear determining equations
vG(x, u, u1, ..., u
p)|G(x,u,u
1,...,u
p)=0 = 0 (2.8)
for m + n unknown functions ξi, ηk that depend on m + n variables (x, u). The determiningequations (2.8) state the invariance of the solution manifold under the action of the vectorfield v.
The overdetermined system for ξi, ηk is obtained from (2.8) using the fact that ξi, ηk do notdepend on derivatives uk
i . Thus the system (2.8) splits into an N ≤ l(mn + 1) linear partialdifferential equations. For example, in the case of the system of adiabatic compressiblePlasma Equilibrium equations, for example, this generally leads to a system of 188 linearPDEs for 13 unknown functions [7].
2. Contact and higher-order symmetries. When tangent vector field components ξi, ηk
of the symmetry generator (2.5) essentially depend on derivatives,
(x′)i = xi + εξi(x, u, u1, ..., u
q) + O(ε2), i = 1, . . . , n,
(u′)j = uj + εηj(x, u, u1, ..., u
q) + O(ε2), j = 1, . . . ,m,
(2.9)
corresponding symmetries are called higher-order symmetries [2]. Unlike Lie symmetries,they act on the solution manifold Ω∞ in the infinite-dimensional space X × U × U1 × · · · ×Up×. . . of all variables and derivatives. Due to this fact, higher-order symmetries, in general,can not be integrated to yield a global group representation similar to (2.2), and can not beused to explicitly map solutions of differential equations into new solutions.
Contact symmetries are symmetries of the form (2.9) that preserve the contact conditiondu′ = u′jdx′j. In the case of one equation, m = 1, tangent vector field components ξi, η ofcontact symmetry generators essentially depend on first derivatives. For m > 1, contactsymmetries are independent of derivatives. Similarly to Lie point symmetries, contact sym-metries act in the finite-dimensional space, and their global group representation can bereconstructed from the local representation.
Similarly to Lie point symmetries, contact and higher-order symmetries are computedalgorithmically – the determining equations are obtained from the requirement (2.8) thatthe given system of equations is invariant under the action of the prolonged vector field(2.5). The overdetermined system of linear PDEs for ξi, ηk is found by equating to zero allcoefficients at derivatives of order higher than q, in the determining equations (2.8).
The implementation of algorithms for finding point and higher-order symmetries in the GeMpackage, as well as examples, are discussed below.
2.2 Methods for finding conservation laws of PDE systems.
Consider the problem of finding conservation laws in the form
DiΦi[u] = 0 (2.10)
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that follow from the system (2.1). The conserved densities (fluxes) Φi[u] may depend on x, uand derivatives up to an arbitrary order.
A conservation law (2.10) is called trivial if its fluxes are
Φi = M i + H i
where M i and H i are sets of smooth functions such that M i vanish on the solutions of thesystem (2.1), and div(H1, ..., Hn) ≡ 0. Two conservation laws DiΦ
i[u] = 0 and DiΨi[u] = 0
are equivalent if Di(Φi[u] − Ψi[u]) = 0 is a trivial conservation law. The more general
‘triviality’ idea is the notion of liner dependence of conservation laws. Conservation laws arelinearly dependent if there exists a linear combination of them which is a trivial conservationlaw.
In practice, it is important to obtain nontrivial linearly independent conservation laws ofa given system of differential equations.
Most of contemporary practically useful methods for finding conservation laws involve con-sidering a linear combination of equations of the given system (2.1) with a set of multipliers(or factors, or characteristic functions) Λp[U ] . Multipliers may depend on independentand dependent variables and derivatives. A linear combination yields a conservation law(2.10) if and only if
Λp[U ]Gp = DiΦi[U ] (2.11)
for some set of fluxes Φi[U ] (here U denotes a vector of arbitrary functions.) Then theconservation law DiΦ
i[u] = 0 holds on on the solutions U = u of the system (2.1).This method does not require the PDE system under consideration to have a variational
formulation, and does not use any version of Noether’s theorem when the considered systemis variational. The method is algorithmic and therefore practical. It yields all conservationlaws with fluxes depending on a prescribed set of dependent and’or independent variablesand derivatives.
We note that any conservation law (2.10) of the system (2.1) may be represented by a linearcombination (2.11), if the given system is not degenerate (namely, is of Cauchy-Kovalevskayatype, see [2].)
The determining equations that yield sets of multipliers Λp[U ] of the conservation lawsfor the given system (2.1) are found from the known observation that an expression is adivergence expression if and only if it is annihilated by Euler operators with respect to alldependent variables in the system. The following theorem holds (see [1, 2]):
Theorem 1. A set of multipliers Λσ[U ] yields a conservation law of the given system ofdifferential equations (2.1) if and only if the equations
EUk(Λi[U ]Gi) = 0, k = 1, . . . , m, (2.12)
where EUk is the Euler operator with respect to Uk:
EUk =∂
∂Uk−Di
∂
∂Uki
+ · · ·+ (−1)jDi1 · · ·Dij
∂
∂Uki1···ij
+ · · · , (2.13)
hold for arbitrary functions U = (U1(x), . . . , Um(x)).
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The equations (2.12) are the linear determining equations for the multipliers for con-servation laws admitted by the given system (2.1). In practice, to perform a computa-tion, one chooses the maximal order of derivatives q ≥ 0 in the dependence of multipliersΛi[U ] = Λi(x, U, U
1, ..., U
q). Then the determining equations (2.12) is split into an overdeter-
mined system of simpler equations, by setting to zero, in all determining equations, of thecoefficients of all derivatives of U higher than q.
After finding a set of multipliers Λσ[U ] that yield a conservation law (2.11), one canobtain, by an algorithmic procedure, the corresponding set of fluxes Φi[u] for each set ofmultipliers. This is achieved either by directly solving the system (2.11) for the fluxes Φi[U ],or using integral formulas arising from homotopy operators ( [17–19].)
Methods involving multipliers yield all conservation laws that involve derivatives up toa certain (prescribed) order. However, in some cases, conservation law analysis throughmultipliers may lead to finding the complete set of conservation laws. A relevant example isthe case of evolution equations of even order, for which the maximal order of conservation lawis finite [32]. Alternatively, if a given system possesses a countable number of conservationlaws, it may be possible to construct them recursively.
The application of GeM symbolic computation routines for generation, splitting and solutionof overdetermined systems that arise in conservation law analysis, and examples conservationlaw analysis of particular equations, is found Section 3.
The use of adjoint linearization to find multipliers for conservation laws is demonstratedin the following subsection. (See [33] and references therein for an extensive discussion.)
2.3 Symmetries, adjoint symmetries and conservation laws of PDEsystems
A symmetry operator (2.4) can be equivalently represented in the evolutionary form (seee.g. [1]):
Xev = ζk[u]∂
∂uk, ζk[u] = ηk(x, u)− uk
i ξi(x, u). (2.14)
The symmetry in the evolutionary form defines a local coordinate transformation
(x′)i = xi, (i = 1, . . . , n),(u′)k = uk + εζk[u] + O(ε2) (k = 1, . . . , m).
From this fact it follows that if L is the linearizing operator (Frechet derivative) for thegiven system (2.1), then the determining equations (2.8) are equivalent to
Lζ[u] = 0. (2.15)
In other words, the operator Xev (2.14) defines a symmetry if and only if its componentssolve the linearized version of the given system (2.1). (The linearization is understood in theFrechet derivative, not about a particular solution.)
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The given system (2.1), as written, has a variational principle (i.e. corresponds to anextremum of some action functional) if and only if L is a self-adjoint operator, i.e., L∗ = Lwhere L∗ is the adjoint of L. A set of functions κ[u] defines an adjoint symmetry if thesefunctions satisfy the adjoint linearized equation
L∗κ[u] = 0. (2.16)
One can show [2] that the set of multipliers Λp[U ] for the conservation law (2.11) satisfiesthe adjoint equations
L∗Λ[u] = 0,
hence a conservation law always defines an adjoint symmetry. However the converse isnot true, since the determining equations (2.12) for conservation law multipliers are morerestrictive than the determining equations for adjoint symmetries (2.16). In particular, theset of determining equations for conservation law multipliers consists of the system of adjointsymmetry determining equations augmented by the extra determining equations [18].
3 ”GeM” symbolic package for general Lie symmetry
computation.
3.1 Description
A Waterloo Maple - based package GeM (”General Module”) has been recently developed bythe author. The package routines are capable of finding local symmetries and prescribedclasses of conservation laws for any ODE/PDE system without significant limitations on DEorder and number of variables, and without human intervention.
The routines of the module allow the analysis of ODE/PDE systems containing arbitraryfunctions. Classification with respect to constitutive functions and isolation cases for whichadditional symmetries / conservation laws occur is automatically accomplished. Classifica-tion problems naturally arise in the analysis of classes of DE systems from applications. Thesymmetry classification may lead to the linearization, discovery of new conservation lawsand symmetries in particular cases (for example, see [22].)
The ”GeM” package [30] employs a special representation of the system under considerationand the determining equations: all dependent variables and derivatives are treated as Maplesymbols, rather than functions or expressions. This significantly speeds up the computationof the tangent vector field components corresponding to derivatives (up to any order) andthe production of the (overdetermined) system of determining equations for the unknowntangent vector field coordinates. The latter is reduced by Rif [31] package routines.
The reduced system of determining equations, in many cases, has a simple form, andcan be integrated by hand. In more complicated situations, it may be automatically andcompletely integrated by Maple’s internal PDSolve routine. However it is known that thisroutine sometimes returns incomplete solutions, therefore its output needs to be tested forcompleteness separately, for instance, using another Maple routine, as shown in Section 3.2.
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A description of symmetry, adjoint symmetry and conservation law analysis proceduresusing the GeM package is given below.
The computation examples are presented in Section 4.
3.2 ”GeM” routine sequence for symmetry analysis
In this section we outline the essential structure of a program for local symmetry analysis ofa PDE system using the GeM package.
1. Initialization.
read(”d:/gem019b.txt”);
2. Declaration of dependent and independent variables, constitutive functions(optional), and free constants (optional).
Variables, free functions and free constants are declared with the following routine:
gem_decl_vars([IV], [DV], [AF], [AC], NDer);
Here [IV] is a vector of independent variables, [DV] a vector of dependent variables, [AF]a vector of arbitrary (constitutive) functions in the system, and [AC] a vector of arbitraryconstants in the system. NDer is the order of highest derivative of a dependent variable onwhich arbitrary functions depend.
3. Declaration of equations of the DE system.
gem_decl_eqs([eqs],[eq_orders],[solve_for]);
Here [eqs] is a vector of differential equations in the form E = 0 or A = B; [eq_orders] avector of orders of differential equations (same sequence as in [eqs] assumed); [solve_for]a vector of quantities (dependent variables or derivatives, usually highest derivatives) theequations are solved for. The differential equations of the system under consideration mustbe solvable for quantities specified in [solve_for]. The simpler the expressions for thesequantities are, the more elegant equations on further steps will look.
Within this routine, all derivatives are defined as Maple symbols: ∂B1/∂x ≡ B1x, etc.For example, the Maple expression for the equation involving a divergence of a 3-vectorB = (B1, B2, B3): div B = 0 is
∂
∂xB1 (x, y, z) +
∂
∂yB2 (x, y, z) +
∂
∂zB3 (x, y, z) = 0
and is represented in GeM by
B1x + B2y + B3z = 0.
After this conversion, the unknown tangent vector field components ξi, ηk are not compositefunctions, but functions of all ”independent variables”, which now include dependent andindependent variables of the system under consideration and all partial derivatives.
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4. Declaration of dependencies of tangent vector field components. The depen-dency list of tangent vector field components ξi, ηk are declared as follows:
gem_def_vf([vars], in_ev_form, max_der_order);
Here [vars] is a vector that may include independent variables, dependent variables, andderivatives of dependent variables.
The flag in_ev_form should have a nonzero value if the symmetries are sought in theevolutionary form (2.14). For the standard form of symmetry generator (2.4), one usesin_ev_form=0.
The parameter max_der_order specifies maximal derivative order on which tangent vectorfield components may depend.
should have a nonzero value if the symmetries are sought in the evolutionary form (2.14).For the standard form of symmetry generator (2.4), one uses in_ev_form=0.
On this stage, one can introduce an ansatz (for example, assume independence of ξi, ηk
on independent variables.) Ansatzes can prove useful for complicated PDE systems wherecomplete symmetry analysis is not affordable.
To find higher-order (and contact, for DEs with one dependent variable) symmetries, oneshould include appropriate derivatives in the [vars] list and use max_der_order>0, whereasfor Lie point symmetries max_der_order=in_ev_form=0.
The function names of resulting tangent vector field components are stored in a global setV_F_COORDS_.
5. Generation of symmetry determining equations and the overdetermined sys-tem. The split system of determining equations is obtained by a command
overdet_sys:=gem_get_split_sys();
Here the split system of determining equations is returned in overdet_sys (and also storedin a global set variable SPLIT_DET_SYS_.) The unsplit determining equations (2.8) are storedin a global variable ALL_XE_wE0, but are seldom used. The equations in SPLIT_DET_SYS_
form an overdetermined system obtained by setting to zero, in all determining equations(2.8), the coefficients of all derivatives uk
i on which tangent vector field components ξi, ηk donot depend.
Overdetermined systems are often large and contain many repeating and dependent equa-tions. In Waterloo Maple simplification can be done, for instance, using the Rif developedby Reid and Wittkopf:
simplified_system:=rifsimp(overdet_sys,convert(V_F_COORDS_,list));
The reader is referred to Maple help on rifsimp for numerous useful options of this com-mand.
The reduced system of determining equations is found in the variable simplified_system[Solved].
Remark 1. In rare cases, the system of determining equations (2.8) can not be splitinto an overdetermined system. In such cases, one applies the rifsimp routine directly todetermining equations stored in a global variable ALL_XE_wE0.
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Remark 2. If the given PDE system contains constitutive function(s), one performs aclassification of all choices of constitutive function(s) that give different results. This isachieved by running the rifsimp command with an option casesplit:
split_system:=rifsimp(overdet_sys, convert(V_F_COORDS_,list), casesplit);
The result is a Maple table of cases (“pivots”) and equations on tangent vector field compo-nents in those cases. A tree of cases can be visualized, with pivots explicitly written, by thecommand
caseplot(split_system, pivots);
6. Solution of determining equations. The possibility of automatic integration of thereduced system of determining equations depends on the actual complexity of the symmetrystructure of the DE system under consideration. In many cases, even ones with rich symmetrystructure (as seen in the example with Plasma Equilibrium equations below), one can findexpressions for the tangent vector field coordinates by hand.
Another option is to use the Maple built-in integrator:
expand(pdsolve(simplified_system[Solved]));
In the case with constitutive function(s), one applies the same routine to one of the splitcases, for example:
expand(pdsolve(split_system[i][Solved]));
where i is an integer between 1 and number of cases.
We remark again that though the Maple pdsolve routine is one of the best (possiblythe best) of its kind, it sometimes returns incomplete or unnecessarily complicated results,or no result, even when is not difficult to obtain it by hand. However we enthusiasticallyrecommend to use it for preliminary analysis.
The completeness of the solution returned by pdsolve can be verified using another routineof the Rif package:
maxdimsystems(overdet_sys, convert(V_F_COORDS_,list));
The result includes the dimension of solution space, which should be compared to the numberof arbitrary constants in the solution returned by pdsolve.
3.3 ”GeM” routine sequence for conservation law / adjoint sym-metry analysis
1-3: Initialization, declaration of variables, functions and equations. Same as forsymmetry analysis.
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4. Generation of the overdetermined PDE system for conservation law multipli-ers / adjoint symmetry components.
The following routine generates an overdetermined system of determining equations forconservation law multipliers / adjoint symmetry components:
CLde:=get_CL_gen_det_eqs([vars],on_sol_space_flag);
The routine proceeds by computing determining equations (2.12) using Euler operators,and setting to zero the coefficients at all derivatives on which the multipliers do not depend.
Here [vars] is a vector that may include independent variables, dependent variables, andderivatives of dependent variables. It specifies the dependency of conservation law multipliersor adjoint symmetry components.
A nonzero value of on_sol_space_flag means that the determining equations (2.12) arecomputed on the solution space of the DE system under consideration, thus adjoint symmetrycomponents are being found. Setting on_sol_space_flag=0 forces the system to work offsolution space and thus seek multipliers of conservation laws.
The set CLde in the above example contains the overdetermined system of conservation lawdetermining equations.
The multipliers (or adjoint symmetry components) Λp[U ] (2.11) defined as functions oftheir arguments are stored in a global variable LAMBDAS_.
The overdetermined system of conservation law determining equations is simplified andreduced:
simplified_system:=rifsimp(CLde,LAMBDAS_);
If the given PDE system contains constitutive function(s), one performs a classificationwith respect to the form of the constitutive function(s) and visualizes the resulting Mapletable of cases (“pivots”) and equations on conservation law multipliers, as above:
split_system:=rifsimp(CLde, LAMBDAS_, casesplit);
caseplot(split_system, pivots);
5. Solution of determining equations. Finding the conservation laws.Computation experience shows that, similarly to the symmetry determining equations, in
many cases, one can find expressions for the tangent vector field coordinates by hand, or byusing the Maple built-in solver
solved_multipliers:=expand(pdsolve(simplified_system[Solved])); (3.1)
In the case with constitutive function(s), one applies the same routine to one of the splitcases, for example:
solved_multipliers:=expand(pdsolve(split_system[i][Solved])); (3.2)
where i is an integer between 1 and the number of cases. The completeness of the returnedsolution is normally checked with the maxdimsystems routine, same way as in Part 6 ofSection 3.2.
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If adjoint symmetry analysis is being undertaken, the procedure stops here since adjointsymmetry components Λp[U ] are found. In the case of conservation law analysis, afterfinding the multipliers Λp[U ], the fluxes Φi[u] = 0 (2.10) of conservation laws are found byconversion of an expression Λp[u]Gp = 0 to the form DiΦ
i[u] = 0, which in many cases is asimple computational exercise. One may use the following routine provided in ”GeM” :
GetFluxes([solved_multipliers_set],[max_der_order]);
As input, this routine takes the set of multipliers found in (3.1) or (3.2). This routine splitsthe multipliers according to free constants present, and, if successful, for each case returnsfluxes of corresponding independent conservation laws.
4 Run examples
This section contains code and output for several examples: symmetry analysis (section4.2),conservation law classification with respect to a constitutive function (section 4.3), andadjoint symmetry analysis (section 4.4).
4.1 Higher-order symmetries of Burgers’ equation
As en example, we present Maple code and output of a program that computes higher-ordersymmetries of Burgers equation. Following [1] (chapter 5.2), we seek symmetries of Burgers’equation
ut + uux = uxx (4.1)
that depend on (x, t, ux, uxx, uxxx), in the evolutionary form (2.14).The Maple program has the form:
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restart; read(”d:/gem019b.txt”);
gem_decl_vars([x,t], [U(x,t)],[ ],[ ],0);
gem_decl_eqs(
[diff(U(x,t),t)+U(x,t)*diff(U(x,t),x) =diff(U(x,t),x,x)],
[2],[diff(U(x,t),t)]);
gem_def_vf([x,t,U(x,t),diff(U(x,t),x),diff(U(x,t),x,x),diff(U(x,t),x,x,x)],1,3);
overdet_sys:=gem_get_split_sys():
split_system:=rifsimp(overdet_sys,convert(V_F_COORDS_,list)):
#Verify dimension of solution space:
maxdimsystems(overdet_sys,convert(V_F_COORDS_,list),output=rif)[dimension];
#Solve for symmetry generator:
SymmetrySolution:=expand(pdsolve(split_system[Solved])):
#Print 9 independent symmetry generators:
GenAllX(SymmetrySolution,9);
The computation of the nine-dimensional Lie algebra of symmetry generators takes severalseconds. The output symmetry generators are shown in Fig. ?? and coincide with the knownresult [1].
4.2 Nonlocal symmetries of the Lagrange subsystem of Planar GasDynamics equations
The Lagrange system of Planar Gas Dynamics (PGD) equations has the form [34]
Ly, s, v, p, q = 0 :
qs − vy = 0,vs + py = 0,ps + B(p, q)vy = 0.
(4.2)
Here the Lagrangian coordinate y essentially enumerates the fluid particles; s is time; ppressure; v velocity; and q = q/ρ inverse density.
In this example, we seek nonlocal symmetries of the Lagrange system of Planar Gas Dy-namics (4.2) by analyzing the symmetries of a nonlocally-related subsystem obtained byexcluding velocity v [22]:
Ly, s, p, q = 0 :
qss + pyy = 0,ps + B(p, q)qs = 0.
(4.3)
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Figure 1: The basis of nine-dimensional Lie algebra of point symmetries of Burgers equation (4.1).
We note that this system and hence nonlocal symmetries resulting from it are not presentin [34].
It is possible to undertake full symmetry classification of the subsystem (4.3) with respectto the constitutive function B(p, q). The classification yields 29 cases and is not includedin this work due to its volume. The program including classification results can be foundat [30].
Here we would like to present a simpler symmetry classification for a family of constitutivefunctions B(p, q) = F (p)/q.
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restart; read(”d:/gem019b.txt”);
B(P(y,s),Q(y,s)):=F(P(y,s))/Q(y,s):
gem_decl_vars([y,s], [Q(y,s),P(y,s)],[F(P(y,s))],[],0);
gem_decl_eqs([
diff(Q(y,s),s,s)+diff(P(y,s),y,y)=0,
diff(P(y,s),s)+B(P(y,s),Q(y,s))*diff(Q(y,s),s)=0 ],
[2,1],[diff(Q(y,s),s,s),diff(P(y,s),s)]);
gem_def_vf([y,s, Q(y,s),P(y,s)],0,0);
overdet_sys:=gem_get_split_sys():
split_system:=rifsimp(overdet_sys,convert(V_F_COORDS_,list)):
#Plot the case tree
caseplot(rss,pivots);
The case tree output by the last command is presented on Fig. 2 and incudes six cases,which can readily be analyzed separately. (Note that though on some systems the numberof cases returned by rifsimp can be different, the classification results remain the same.)
As an example, we analyze case 6: F (p) = −p, which corresponds to Chaplygin gas:B(p, q) = −p/q. The above program is appended by
chapl_case:=split_system[6][Solved];
#Check dimension:
maxdimsystems(chapl_case,convert(V_F_COORDS_,list),output=rif)[dimension];
#compute 10-dimensional symmetry generator components:
SymmetrySolution:=pdsolve(subs(F(P)=-P,chapl_case),convert(V_F_COORDS_,list));
#Print all symmetry generators:
GenAllX(SymmetrySolution,10);
The output of the last command is presented on Fig. 3. The system equation (4.3) forChaplygin gas is thus found to possess a ten-dimensional Lie algebra of point symmetries. Inparticular, generators X1, X3, X6 define nonlocal symmetries of the Lagrange system (4.2).(This is found from the comparison with the basis of symmetry generators for the Lagrangesystem (4.3), see, e.g., [34].)
4.3 Conservation law classification for the three-dimensional Cahn-Hilliard equation
An important problem in the conservation law analysis is the classification of conservationlaws in a problem (or rather a family of problems) containing a constitutive function (seeSection 2.2.)
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Figure 2: The symmetry classification case for the subsystem (4.3) and the constitutive functionB(p, q) = F (p)/q.
In this example, we present new results of the conservation law classification of the three-dimensional Cahn-Hilliard equation. To the best of author’s knowledge, this analysis hasnot appeared in literature before.
The Cahn-Hilliard equation (4.4) is a model for the dynamics of phase separation in binaryalloys [35]. It is given by
ut = ∆(−a2∆u−Q(u)), (4.4)
where ∆ = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 is the 3D Laplacian; u = u(t, x, y, z) the dependentvariable (concentration of one of the alloy components); x, y, z spatial variables; t time; Q(u)a constitutive function related to the diffusion coefficient; and a 6= 0 a parameter.
We look for conservation laws of this PDE of order four, with four independent variables,according to the procedure described in Section 2.2. There is only one equation and hanceone multiplier in this problem; the ansatz for the multiplier is chosen to be of order zero:Λ = Λ(t, x, y, z, u).
The Maple program is given below, with some output omitted in order to save space. It
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Figure 3: The ten-dimensional Lie algebra of point symmetries of the subsystem (4.3).
can be downloaded from [30].
restart; read(”d:/gem019b.txt”);
gem_decl_vars([t,x,y,z], [U(x,y,z,t)],[Q(U(x,y,z,t))],[a],0);
gem_decl_eqs([
diff(U(x,y,z,t),t)=laplacian( -a^2*laplacian(U(x,y,z,t),[x,y,z]) - Q(U(x,y,z,t)),
[x,y,z])],
[4],[diff(U(x,y,z,t),x,x,x,x)]);
#Generate and print conservation law determining equations:
CLde:=get_CL_gen_det_eqs([x,y,z,t,U(x,y,z,t)],0,0);
#Now the system must be studied for different cases of ’Q(u)’
split_system:=rifsimp(CLde,convert(LAMBDAS_,list),casesplit);
#Display the case tree and cases (pivots):
caseplot(split_system,pivots);
Evidently the only nonlinear case is Case 1, Q′′(u) 6= 0, a 6= 0, for which we have the system
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Figure 4: The case tree and pivots for the problem of conservation law analysis of the 3D Cahn-Hilliard equation (4.4).
of equations on the multiplier shown by
split_system[1][Solved];
The output equations are
[Λx,x + Λy,y + Λz,z = 0, Λt = 0, ΛU = 0]
where, as in Fig. 4 above, subscripts denote partial derivatives. Hence any functionΛ1(x, y, z) harmonic in 3D space is a multiplier that produces a nontrivial conservationlaw of the 3D Cahn-Hilliard equation (4.4).
Remark 1. By a straightforward computation, one is able to generalize the result: ann−dimensional Cahn-Hilliard equation ut = ∆(−a2∆u−Q(u)), ∆ being an n−dimensionalLaplacian, admits nontrivial conservation laws with multipliers satisfying ∆Λ(x1, . . . , xn) =0.
Remark 2. The computation results thus show that the set of conservation law multipliersthat depend only on the dependent variable u and independent variables, for Cahn-Hilliardequations in two– and three–dimensional space, is exhausted by harmonic functions in thecorresponding space, which depend only on spatial variables. For the case of one spacedimension, the set of conservation law multipliers that depend on the dependent variable u,independent variables, and partial derivatives up to order three, is exhausted by functionsΛ(x, t, u, u
1, u
2, u
3) = c1x + c2, c1, c2 = const.
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4.4 Local conservation laws of Nonlinear Telegraph equations
Consider Nonlinear Telegraph (NLT) equations
utt − (F (u)ux)x − (G(u))x = 0. (4.5)
with power nonlinearities F (u) = uα, G(u) = uβ (α, β 6= 0).We perform a classification of conservation laws of these equations using the GeM imple-
mentation of the algorithm described in Section 2.2. We find all conservation laws withmultipliers of the form Λ = Λ(x, t, U).
The Maple program is
restart; read(”d:/gem019b.txt”);
gem_decl_vars([x,t], [U(x,t)],[F(U(x,t)),G(U(x,t))],[ ],0);
gem_decl_eqs([
diff(U(x,t),t,t)-diff( (F(U(x,t))*diff(U(x,t),x) + G(U(x,t))) ,x)=0]
[2],[diff(U(x,t),t,t)]);
#Generate and print conservation law determining equations.
CLde:=get_CL_gen_det_eqs([x,t,U(x,t)],0,0);
#Simplify and split cases
split_system:=rifsimp(CLde union F(U)<>0,LAMBDAS_,casesplit);
caseplot(split_system,pivots);
The output of the last command, showing the tree of cases in the conservation law classi-fication, is shown on Fig. 5.
We now explicitly find conservation laws for all cases, restricting ourselves to power non-linearities F (u) = uα, G(u) = uβ (α, β 6= 0).
The most general Case 1 holds for arbitrary α, β 6= 0. We compute and output conservationlaws with the following commands: The Maple program is
mm:=1;
maxdimsystems(split_system[mm][Solved],LAMBDAS_);
lambdas_solved:=pdsolve(split_system[mm][Solved]);
GetFluxes(lambdas_solved,0);
The following multipliers conservation laws are found for the general Case 1 (Note that theyhold not only for power nonlinearities F (u), G(u) but for arbitrary ones):
=== The Maple program that incorporates the complete analysis performed in this Sub-section is found in [30].
4.5 An adjoint symmetry computation
In this section, we follow [20] to compare symmetries and adjoint symmetries (in evolutionaryform) of a nonlinear PDE equation
utt + H ′(ux)uxx + H(ux) = 0, (4.6)
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Figure 5: Tree of cases of constitutive functions F (u), G(u) in the classification of conservationlaws of the NLT equations (4.5)
for the case H(y) = y4.The following program computes symmetries of (4.6) in evolutionary form (note the argu-
ments (..., 1, 1) in the call to gem_def_vf):
restart; read(”d:/gem019b.txt”);
gem_decl_vars([t,x], [U(x,t)],[ ],[ ],0);
H:=y->y^4;
gem_decl_eqs([
exp(x)*(diff(U(x,t),t,t)+diff(H(diff(U(x,t),x)),x)+H(diff(U(x,t),x)))=0]
[2],[diff(U(x,t),t,t)]);
gem_def_vf([x,t, U(x,t),diff(U(x,t),x),diff(U(x,t),t)],1,1);
overdet_sys:=gem_get_split_sys():
sym_tangent_v_field:=pdsolve(split_system[Solved]);
#Print the general symmetry generators:
GenX(sym_tangent_v_field);
The output symmetry generator is
X =
(C1 + C2t + C3ux + C4ut + C5
(tut +
2
3u
))∂
∂u.
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We now compute adjoint symmetries of (4.6):
restart; read(”d:/gem019b.txt”);
gem_decl_vars([t,x], [U(x,t)],[ ],[ ],0);
H:=y->y^4;
gem_decl_eqs([
exp(x)*(diff(U(x,t),t,t)+diff(H(diff(U(x,t),x)),x)+H(diff(U(x,t),x)))=0]
[2],[diff(U(x,t),t,t)]);
CLde:=get_CL_gen_det_eqs([t,x,U(x,t),diff(U(x,t),x),diff(U(x,t),t)],1,1);
rss:=rifsimp(CLde,convert(LAMBDAS_,list));
sym_tangent_v_field:=pdsolve(split_system[Solved]);
adjsym_tangent_v_field:=pdsolve(rss[Solved]);
The resulting adjoint symmetry generator is
X∗ = ex
(C1 + C2t + C3ux + C4ut + C5
(tut +
2
3u
))∂
∂u,
which does not coincide with X above due to the factor ex. Hence the system (4.6) is notself-adjoint as it stands, and hence does not follow from a variational principle.
However, an analogous computation shows, as noted in [20], for the equation
ex (utt + H ′(ux)uxx + H(ux)) = 0, (4.7)
its linearization and adjoint linear system coincide, (hence point symmetry and adjoint sym-metry generators also coincide), and therefore (4.7) does follow from a variational principle.This example shows that the property of existence of a variational formulation for a nonlinearPDE is not ”invariant” in a reasonable sense, and therefore artificial.
5 Summary and further remarks
The ”GeM” module contains a set of handy automated tools for local (Lie, contact andhigher-order) symmetry analysis, conservation law analysis (through the determination ofconservation law multipliers), and adjoint symmetry analysis of systems of partial differ-ential equations. ”GeM” is successfully used to solve problems involving classification ofsymmetries and conservation laws of systems containing arbitrary (constitutive) functions.
The module has shown reliable behaviour in problems involving PDEs with several depen-dent / independent variables (such as equations of incompressible MagnetohydrodynamicEquilibrium equations with 3 independent and 8 dependent variables), and nonlinear DEs ofhigher order (see the example with Cahn-Hilliard equation of order four, with one dependentand four independent variables: Section 4.3).
Further development will proceed in the following directions:(1.) Increasing work efficiency and speed in problems that involve more than two indepen-
dent variables and (explicitly or implicitly) derivatives of high orders, 8 and above;
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(2.) Increasing the effectiveness of integration of expressions of the type M = Λp[U ]Gp = 0to obtain the canonical form of the corresponding conservation law DiΦ
i[U ] = 0, by usingalternative methods (including homotopy operators);
(3.) Automatic generation, on demand, of all potential systems of next level, based onavailable conservation laws of the given system (see [22]), for the construction of trees ofnonlocally related potential systems. Automatic generation of nonlocally related subsystems.
AcknowledgementsThe author thanks George Bluman, Stephen Anco and Nataliya Ivanova for discussion and
suggestions.
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