symbolic computation of conservation laws for nonlinear
TRANSCRIPT
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Symbolic computation of conservation laws for nonlinear partial
differential equations in multiple space dimensions
Douglas Poole and Willy Hereman
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401, USA
Abstract
A method for symbolically computing conservation laws of nonlinear partial differentialequations (PDEs) in multiple space dimensions is presented in the language of variationalcalculus and linear algebra. The steps of the method are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili equations as examples.
The method is algorithmic and has been implemented in Mathematica. The softwarepackage, ConservationLawsMD.m, can be used to symbolically compute and test con-servation laws for polynomial PDEs that can be written as nonlinear evolution equations.
The code ConservationLawsMD.m has been applied to (2+1)-dimensional versionsof the Sawada-Kotera, Camassa-Holm, and Gardner equations, and the multi-dimensionalKhokhlov-Zabolotskaya equation.
Keywords: Conservation laws; Nonlinear PDEs; Symbolic software; Complete integrability
1. Introduction
Many nonlinear partial differential equations (PDEs) in the applied sciences and engi-neering are continuity equations which express conservation of mass, momentum, energy, orelectric charge. Such equations occur in, e.g., fluid mechanics, particle and quantum physics,plasma physics, elasticity, gas dynamics, electromagnetism, magneto-hydro-dynamics, non-linear optics, etc. Certain nonlinear PDEs admit infinitely many conservation laws. Al-though most lack a physical interpretation, these conservation laws play an important rolein establishing the complete integrability of the PDE. Completely integrable PDEs are non-linear PDEs that can be linearized by some transformation or explicitly solved with theInverse Scattering Transform (IST). See, e.g., Ablowitz and Clarkson (1991).
The search for conservation laws of the Korteweg-de Vries (KdV) equation began around1964 and the knowledge of conservation laws was paramount for the development of soliton
Email address: [email protected] and [email protected] (Douglas Poole and Willy Hereman)URL: http://inside.mines.edu/~whereman (Douglas Poole and Willy Hereman)
Preprint submitted to Elsevier June 21, 2010
theory. As Newell (1983) narrates, it led to the discovery of the Miura transformation(which connects solutions of the KdV and modified KdV equations) and the Lax pair forthe KdV equation. A Lax pair (Lax, 1968) is a set of two linear equations. The compatibilityconditions of these equations is the nonlinear PDE. The Lax pair is the starting point forthe IST (Ablowitz and Clarkson, 1991; Ablowitz and Segur, 1981). which has been used toconstruct soliton solutions, i.e., stable solutions that interact elastically upon collision.
Conversely, the existence of many (independent) conserved densities is a predictor forcomplete integrability. The knowledge of conservation laws also aids the study of qualita-tive properties of PDEs, in particular, bi-Hamiltonian structures and recursion operators(Baldwin and Hereman, 2010). Furthermore, if constitutive properties have been added to“close a model,” one should verify that conserved quantities have remained intact. Anotherapplication involves numerical solvers for PDEs (Sanz-Serna, 1982), where one checks if thefirst few (discretized) conserved densities are preserved after each time step.
There are several methods for computing conservation laws as discussed by Bluman et al.(2010), Hereman et al. (2005), Naz (2008), and Naz et al. (2008). One could apply Noether’stheorem, which states that a (variational) symmetry of the PDE corresponds to a conser-vation law. Using Noether’s method, the differential geometry package in Maple containstools for conservation laws developed by Anderson (2004a) and Anderson and Cheb-Terrab(2009). Circumventing Noether’s Theorem, Wolf (2002) has developed four programs inREDUCE which solve an over-determined system of differential equations to get conserva-tion laws. Based on the integrating factor method, Cheviakov (2007, 2010) has written aMaple program that computes a set of multipliers on the PDE. To find conservation laws,here again, one has to solve a system of differential equations. Last, conservation laws canbe obtained from the Lax operators, as shown in, e.g., Drinfel’d and Sokolov (1985).
By contrast, the direct method discussed in this paper uses tools from calculus, the cal-culus of variations, linear algebra, and differential geometry. Briefly, our method works asfollows. A candidate (local) density is assumed to be a linear combination with undeter-mined coefficients of monomials that are invariant under the scaling symmetry of the PDE.Next, the time derivative of the candidate density is computed and evaluated on the PDE.Subsequently, the variational derivative is applied to get a linear system for the undeter-mined coefficients. The solution of that system is substituted into the candidate density.Once the density is known, the flux is obtained by applying a homotopy operator to invert adivergence. Our method can be implemented in any major computer algebra system (CAS).The package ConservationLawsMD.m by Poole and Hereman (2009) is a Mathematicaimplementation based on Hereman et al. (2005), with new features added by Poole (2009).
This paper is organized as follows. Section 2 shows conservation laws for the Zakharov-Kuznetsov (ZK) and Kadomtsev-Petviashvili (KP) equations. Section 3 covers the toolsthat will be used in the algorithm. In Section 4, the algorithm is presented and illustratedfor the ZK and KP equations. Section 6 discusses conservation laws of PDEs in multi-ple space dimensions, including the Sawada-Kotera, Khokhlov-Zabolotskaya, and Camassa-Holm equations. A general conservation law for the KP equation is given in Section 5.Using the (2+1)-dimensional Gardner equation as an example, Section 7 shows how to useConservationLawsMD.m. Finally, some conclusions are drawn in Section 8.
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2. Examples of Conservation Laws
This paper deals with systems of PDEs of order P ,
∆(u(P )(x, t)) = 0, (1)
in (n + 1) dimensions where n refers to the number of space variables, i.e., the numberof components of x. The additional independent variable is time (t). u(P )(x, t) denotes thedependent variable u = (u1, . . . , uj, . . . , uN) and its partial derivatives (up to order M) withrespect to x and t.
Although the method discussed in this paper works for arbitrary n, the algorithm andcode are restricted to 1 D, 2 D, and 3 D cases. Hence, the space variable x represents x, (x, y),or (x, y, z), respectively. For simplicity, in the examples we will denote u1, u2, u3, etc. byu, v, w, etc. Partial derivatives are denoted by subscripts, e.g., ∂5u
∂x3∂y2is written as u3x 2y.
A conservation law for (1) is a scalar PDE in the form
Dtρ+ Div J = 0 on ∆ = 0, (2)
where ρ = ρ(x, t,u(M)(x, t)) is the conserved density of order M , and J = J(x, t,u(K)(x, t)
)is the associated flux of order K. Equation (2) is satisfied for all solutions of (1). In (2),Div J = DxJ1 +DyJ2 if J = (J1, J2) and Div J = DxJ1 +DyJ2 +DzJ3 if J = (J1, J2, J3).
Logically, Dt, Dx, Dy, andDz are total derivative operators and Div is the total divergenceoperator in 2D or 3D, respectively. For example, the total derivative operator Dx (in 1 D)acting on f = f(x, t,u(M)(x, t)) of order M is defined as
Dxf =∂f
∂x+
N∑j=1
Mj1∑
k=0
uj(k+1)x
∂f
∂ujkx, (3)
where M j1 is the order of f in component uj and M = maxj=1,...,NM
j1 . The partial derivative
∂∂x
acts on any x that appears explicitly in f , but not on uj or any partial derivatives of uj.Total derivative operators in multiple dimensions are defined analogously (see Section 3).
The algorithm described in Section 4 allows one to compute local conservation laws forsystems of PDEs that can be written as nonlinear evolution equations, say, in time t,
ut = G(x,u(M)(x)), (4)
where G is assumed to be smooth.Not all multi-dimensional systems of PDEs are evolution equations of type (4). However,
the (3+1)-dimensional continuity equation (2), Dtρ+DxJ1 +DyJ2 +DzJ3 = 0, can be takenas, e.g., DxJ1 + Div(ρ, J2, J3), making it possible to first compute J1, the x-component ofthe flux, then use J1 to compute the rest of the conservation law. Thus, if a PDE can betransformed into an evolution equation in one of the space variables, the algorithm will stillwork. For efficiency, our algorithm does an internal interchange of variables so that theequations adhere to (4). However, the interchange is not used in this paper to keep thedescription of the algorithm clear.
We now introduce two well-documented PDEs together with some of their conservationlaws. These PDEs will also be used in Section 4 to illustrate the steps of the algorithm.
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Example 1. The Zakharov-Kuznetsov (ZK) equation is an evolution equation that char-acterizes three-dimensional ion-sound solitons in a low pressure uniform magnetized plasma(Zakharov and Kuznetsov, 1974). After re-scaling, it has the form
ut + αuux + β∆ux = 0, (5)
where u(x, y, z, t), α and β are real parameters, and ∆ = ∂2
∂x2 + ∂2
∂y2+ ∂2
∂z2is the Laplacian in
three dimensions. The conservation laws for the (2+1)-dimensional ZK equation,
ut + αuux + β(u2x + u2y)x = 0, (6)
where u(x, y, t), were studied by, e.g., Zakharov and Kuznetsov (1974), Infeld (1985), andShivamoggi et al. (1993). Summarizing their results,
Dt(u)+Dx(α2u2 + βu2x)+Dy(βuxy) = 0, (7)
Dt(u2)
+Dx(
2α3u3 − β(u2
x − u2y) + 2βu(u2x + u2y)
)−Dy
(2βuxuy
)= 0, (8)
Dt(u3 − 3β
α(u2
x + u2y))
+Dx(
3α4u4 + 3βu2u2x − 6βu(u2
x + u2y) + 3β2
α(u2
2x − u22y)
− 6β2
α(ux(u3x + ux2y) + uy(u2xy + u3y))
)+Dy
(3βu2uxy + 6β2
αuxy(u2x + u2y)
)= 0, (9)
Dt(tu2− 2
αxu)
+Dx(t(2α
3u3−β(u2
x−u2y)+2βu(u2x+u2y))− 2
αx(α
2u2+βu2x)+ 2β
αux
)−Dy
(2β(tuxuy+ 1
αxuxy)
)= 0. (10)
Example 2. The well-known (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation,
(ut + αuux + u3x)x + σ2u2y = 0, (11)
for u(x, y, t), describes shallow water waves with wavelengths much greater than their ampli-tude moving in the x-direction and subject to weak variations in the y-direction (Kadomtsevand Petviashvili, 1970). The parameter α occurs after a re-scaling of the physical coefficientsand σ2 = ±1. The KP equation is not an evolution equation. However, it can be written asan evolution system in the space variable y,
uy = v, vy = −σ2(utx + αu2x + αuu2x + u4x). (12)
Note that 1σ2 = σ2, and thus σ4 = 1. System (12) instead of (11) will be used in the algorithm
in Section 4.Equation (11) expresses conservation of momentum:
Dt(ux) +Dx(αuux + u3x) +Dy(σ2uy) = 0. (13)
Other well-documented conservation laws (Wolf, 2002) are
Dt(fu)
+Dx(f(1
2αu2 + u2x) + (1
2σ2f ′y2 − fx)(ut + αuux + u3x)
)−Dy
(f ′yu− uy(1
2f ′y2 − σ2fx)
)= 0, (14)
Dt(fyu
)+Dx
(fy(1
2αu2 + u2x) + (1
6σ2f ′y3 − fxy)(ut + αuux − u3x)
)−Dy
(u(1
2f ′y2 − σ2fx)− uy(1
6f ′y3 − σ2fxy)
)= 0, (15)
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where f = f(t) is an arbitrary function. The arbitrary coefficient f leads to an infinitefamily of conservation laws, each of the form (14) or (15). In Section 4 we will show how(7)-(10), (14) and (15) are computed with our algorithm.
3. Tools from the Calculus of Variations and Differential Geometry
Operations are carried out on differential functions, f(x, t,u(M)(x)), on the jet space.Assuming that all partial derivatives of u with respect to t are eliminated from f (viz., t isa parameter), then
u(M)(x) = (u1, u1x, u
1y, u
1z, u
12x, u
12y, u
12z, u
1xy, . . . , u
NMN
1 xMN2 yMN
3 z), (16)
where M j1 , M j
2 , and M j3 are the orders of component uj with respect to x, y, and z, respec-
tively, and M is the maximum total order of all terms in the differential function.Derivatives on the jet space are called total derivatives; an example was given in (3).
By contrast, partial derivatives like ∂k1+k2+k3u∂xk1∂yk2∂zk3
are denoted by uk1x k2y k3z. Although variablecoefficients are allowed, the differential functions should not contain terms that are functionsof (x, t) only (i.e., without being the coefficient of some variable in the jet space).
Three operators for the calculus of variations and differential geometry will play a majorrole in the conservation law algorithm. They are: the total derivative, the Euler operator(also known as the variational derivative), and the homotopy operator. All three operators(which act on the jet space) can be defined algorithmically which allows for straightforwardand efficient computations.
Definition 1. The total derivative operator Dx (in 2 D), acting on f = f(x, y, t,u(M)(x, y)),is defined as
Dxf =∂f
∂x+
N∑j=1
Mj1∑
k1=0
Mj2∑
k2=0
uj(k1+1)x k2y
∂f
∂ujk1x k2y, (17)
where M j1 and M j
2 are the orders of f for component uj with respect to x and y, respectively.Dt, Dy,Dz can be defined analogously.
If the total derivative operator were applied by hand to a differential function f(x, t,u(M)(x)),one would use the product and chain rules to complete the computation. However, formulaslike (3) and (17) are more suitable for symbolic computation.
The Euler operator, which plays a fundamental role in the calculus of variations (Olver,1993), allows one to test if differential functions are exact. Testing exactness is a key stepin the computation of conservation laws.
Definition 2. The Euler operator for the 1 D case where f = f(x,u(M)(x)) is defined as
Luj(x)f =
Mj1∑
k=0
(−Dx)k∂f
∂ujkx
=∂f
∂uj−Dx
∂f
∂ujx+D2
x
∂f
∂uj2x−D3
x
∂f
∂uj3x+ · · ·+ (−Dx)M
j1
∂f
∂ujMj
1x
, (18)
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j = 1, . . . , N. The 2 D and 3 D Euler operators (variational derivatives) are defined analo-gously (Olver, 1993). For example, in the 2 D case where x = (x, y) and f = f(x,u(M)(x)),
Luj(x,y)f =
Mj1∑
k1=0
Mj2∑
k2=0
(−Dx)k1(−Dy)k2∂f
∂uk1x k2y, j = 1, . . . , N. (19)
We suppressed the parameter t in (17)-(19). As we will see later on, in some applications itwill be necessary to swap the meaning of t and y (so that the latter becomes the parameter).Obviously, application of the 2 D Euler operator to f = f(t, x,u(M)(t, x)) instead of f =f(x, y,u(M)(x, y)) would then be accomplished by using Luj(t,x), defined analogously to (19).This remark applies to all operators used in this paper. The following theorem relates to anapplication of the Euler operator.
Definition 3. Let f = f(x,u(M)(x)) be a differential function of order M. When x = x,f is called exact if there exists a differential function F (x,u(M−1)(x)) such that f = DxF .When x = (x, y) or x = (x, y, z), f is exact if there exists a differential vector functionF(x,u(M−1)(x)) such that f = Div F.
Theorem 1. A differential function f = f(x,u(M)(x)) is exact if and only if Lu(x)f ≡ 0.Here, 0 is the vector (0, 0, · · · , 0) which has N components matching the number of compo-nents of u.
Proof. The proof for a general multi-dimensional case is given in, e.g., Poole (2009).
The homotopy operator (Anderson, 2004b; Olver, 1993) integrates exact 1 D differentialfunctions, or inverts the total divergence of exact 2 D or 3 D differential functions. Integra-tion routines of CAS have been unreliable when integrating exact differential expressionsinvolving unspecified functions. Often the built-in integration by parts routines fail whenarbitrary functions appear in the integrand. The 1 D homotopy operator offers an attractivealternative to integration since it circumvents integration by parts altogether.
Definition 4. Let x = x be the independent variable and f = f(x,u(M)(x)) be an exactdifferential function, i.e., there exists a function F such that F = D−1
x f. Thus, F is theintegral of f. The 1 D homotopy operator is defined as
Hu(x)f =
∫ 1
0
(N∑j=1
Iuj(x)f
)[λu]
dλ
λ, (20)
where u = (u1, . . . , uj, . . . , uN). The integrand, Iuj(x)f, is defined as
Iuj(x)f =
Mj1∑
k=1
(k−1∑i=0
ujix (−Dx)k−(i+1)
)∂f
∂ujkx, (21)
where M j1 is the order of f in dependent variable uj with respect to x. The notation f [λu]
means that in f one replaces u by λu, ux by λux, and so on for all derivatives of u, whereλ is an auxiliary parameter.
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Given an exact differential function, the 1 D homotopy operator (20) replaces integrationby parts (in x) with a sequence of differentiations followed by a standard integration withrespect to λ. Indeed, the following theorem states one purpose of the homotopy operator.
Theorem 2. Let f = f(x,u(M)(x)) be exact, i.e., DxF = f for some differential functionF (x,u(M−1)(x)). Then, F = D−1
x f = Hu(x)f.
Proof. A proof for the 1 D case in the language of standard calculus is given in Poole andHereman (2010). Olver (1993) gives a proof based on the variational complex.
The homotopy operator (20) has been a reliable tool for integrating exact polynomial dif-ferential expressions. For applications, see Cheviakov (2007, 2010); Deconinck and Nivala(2009); Hereman (2006); Hereman et al. (2007). However, the homotopy operator fails tointegrate certain classes of exact rational expressions as discussed in Poole and Hereman(2010). We will not consider rational expressions in this paper. Yet, the homotopy integra-tor code in Poole and Hereman (2009) covers large classes of exact rational functions.
CAS cannot invert the divergences of exact 2 D and 3 D differential functions. The 2 Dand 3 D homotopy operators are valuable tools for these tasks.
Definition 5. Let f(x,u(M)(x)) be an exact differential function involving two independentvariables x = (x, y). The 2 D homotopy operator is a “vector” operator with two components,(
H(x)u(x,y)f,H
(y)u(x,y)f
), (22)
where
H(x)u(x,y)f =
∫ 1
0
(N∑j=1
I(x)
uj(x,y)f
)[λu]
dλ
λand H(y)
u(x,y)f =
∫ 1
0
(N∑j=1
I(y)
uj(x,y)f
)[λu]
dλ
λ. (23)
The x-integrand, I(x)
uj(x,y)f , is given by
I(x)
uj(x,y)f =
Mj1∑
k1=1
Mj2∑
k2=0
(k1−1∑i1=0
k2∑i2=0
B(x) uji1x i2y (−Dx)k1−i1−1(−Dy)k2−i2)
∂f
∂ujk1x k2y, (24)
with combinatorial coefficient B(x) = B(i1, i2, k1, k2) defined as
B(i1, i2, k1, k2) =
(i1+i2i1
)(k1+k2−i1−i2−1
k1−i1−1
)(k1+k2k1
) . (25)
Similarly, the y-integrand, I(y)
uj(x,y)f, is defined as
I(y)
uj(x,y)f =
Mj1∑
k1=0
Mj2∑
k2=1
(k1∑i1=0
k2−1∑i2=0
B(y) uji1x i2y (−Dx)k1−i1(−Dy)k2−i2−1
)∂f
∂ujk1x k2y, (26)
where B(y) = B(i2, i1, k2, k1).
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Definition 6. Let f(x,u(M)(x)) be a differential function of three independent variablesx = (x, y, z). The homotopy operator in 3 D is a three-component vector operator,(
H(x)u(x,y,z)f,H
(y)u(x,y,z)f,H
(z)u(x,y,z)f
), (27)
where the x-component is given by
H(x)u(x,y,z)f =
∫ 1
0
(N∑j=1
I(x)
uj(x,y,z)f
)[λu]
dλ
λ. (28)
The y- and z-components are defined analogously. The x-integrand is given by
I(x)
uj(x,y,z)f =
Mj1∑
k1=1
Mj2∑
k2=0
Mj3∑
k3=0
k1−1∑i1=0
k2∑i2=0
k3∑i3=0
(B(x) uji1x i2y i3z
(−Dx)k1−i1−1 (−Dy)k2−i2 (−Dz)k3−i3) ∂f
∂ujk1x k2y k3z, (29)
with combinatorial coefficient B(x) = B(i1, i2, i3, k1, k2, k3) defined as
B(i1, i2, i3, k1, k2, k3) =
(i1+i2+i3
i1
) (i2+i3i2
) (k1+k2+k3−i1−i2−i3−1
k1−i1−1
) (k2+k3−i2−i3
k2−i2
)(k1+k2+k3
k1
) (k2+k3k2
) . (30)
The integrands I(y)
uj(x,y,z)f and I
(z)
uj(x,y,z)f are defined analogously. Based on cyclic per-
mutations, they have combinatorial coefficients B(y) = B(i2, i3, i1, k2, k3, k1) and B(z) =B(i3, i1, i2, k3, k1, k2), respectively.
Using homotopy operators, Div−1 is guaranteed by the following theorem.
Theorem 3. Let f = f(x,u(M)(x)) be exact, i.e., f = Div F for some F(x,u(M−1)(x)).
Then, in the 2 D case, F = Div−1f =(H(x)
u(x,y)f,H(y)u(x,y)f
). Analogously, in 3 D, F =
Div−1f =(H(x)
u(x,y,z)f,H(y)u(x,y,z)f,H
(z)u(x,y,z)f
).
Proof. A proof for the 2 D case is given in Poole (2009). The 3 D case could be provenwith similar arguments.
Unfortunately, the outcome of the homotopy operator is not unique. The integral in the 1 Dcase has a harmless arbitrary constant. However, in the 2 D and 3 D cases there are infinitelymany non-trivial choices for F. From vector calculus we know that Div Curl K = 0. Thus,the addition of Curl K to F would not alter Div F. More precisely, for K = (Dyθ,−Dxθ) in2 D, or for K = (Dyη −Dzξ,Dzθ −Dxη,Dxξ −Dyθ) in 3 D, Div G = Div (F + K) = Div F,where θ, η, and ξ are arbitrary functions. To obtain a concise result for Div−1, Poole andHereman (2010) developed an algorithm that removes curl terms.
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4. An Algorithm for Computing a Conservation Law
To compute a conservation law we construct a candidate density by taking a linearcombination (with undetermined coefficients) of terms that are invariant under the scalingsymmetry of the PDE. The total time derivative of the candidate is computed and evaluatedon (4), thus removing all time derivatives from the problem. The resulting expression mustbe exact. Thus, we use Theorem 1 from Section 3 to derive the linear system that gives theundetermined coefficients. Substituting the coefficients into the candidate leads to a validdensity. To compute the associated flux we invert the divergence with the homotopy oper-ator. To illustrate the subtleties of the algorithm we intersperse the steps of the algorithmwith two examples, viz., the ZK and KP equations.
4.1. Computing the Scaling Symmetry
A PDE has a unique set of Lie-point symmetries which may include translations, ro-tations, dilations, and other symmetries (Bluman et al., 2010). The application of suchsymmetries allows one to generate new solutions from known solutions. We will use onlyone type of Lie-point symmetry, namely, the scaling or dilation symmetry, to formulate a“candidate density.”
Let us assume that a PDE has a scaling symmetry. For example, the ZK equation (6) isinvariant under the scaling symmetry
(x, y, t, u)→ (λ−1x, λ−1y, λ−3t, λ2u), (31)
where λ is an arbitrary scaling parameter. To compute (31) with linear algebra, each variableis assigned an unknown weight.
Definition 7. The weight of a variable is defined as the exponent p in the factor λp thatmultiplies the variable. For the scaling symmetry x→ λ−px, the weight is denoted W (x) =−p. Total derivatives carry a weight. Indeed, if W (x) = −p, then W (Dx) = p.
When a PDE is invariant under a scaling symmetry, we say that it is uniform in rank.
Definition 8. The rank of a monomial is the sum of the weights of the variables in themonomial. A differential function is uniform in rank if all monomials in the differentialfunction have the same rank. A differential function is multi-uniform in rank if it is uniformin rank for more than one scaling symmetry.
Example 3. The term αuux in the ZK equation (6) has rank W (u)+W (u)+W (Dx). Using(31), W (u) + W (u) + W (Dx) = 5. The ZK equation is uniform in rank (invariant undera scaling symmetry) since all other terms have a rank of 5. Note that the parameter α isassumed to have zero weight.
Step 1-ZK (Computing the scaling symmetry). We assume that the PDE is uniform in rank.Under that assumption, we can form a system of weight-balance equations corresponding tothe terms in the PDE. The solution of that system determines the scaling symmetry.
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The weight-balance equations for the ZK equation (6) are
W (u) +W (Dt) = 2W (u) +W (Dx) = W (u) + 3W (Dx) = W (u) +W (Dx) + 2W (Dy). (32)
The parameters α and β are assumed to have zero weight, hence, they are not included inthe system. Solving the linear system gives
W (u) = 2W (Dx), W (Dt) = 3W (Dx), W (Dy) = W (Dx). (33)
To get (31), set W (Dx) = 1. The solution to the weight-balance system is then
W (u) = 2, W (Dt) = 3, W (Dx) = 1, W (Dy) = 1. (34)
Since λ is arbitrary, the weight of at least one of the variables will remain undetermined.We express the weights of the dependent variables in terms of those of the independentvariables, and then assign a value(s) to the arbitrary weight(s). The choice, W (Dx) = 1 in(34) is convenient, however, all undetermined weights must be chosen in such a way that allweights are positive.
If the system of weight-balance equations has no solution, or requires variables to havea zero weight, then the PDE does not have a scaling symmetry. As shown in Section 6.3,it is possible to induce a scaling symmetry by either giving weights to existing parametersin the PDE, or by multiplying some terms in the PDE by weighted parameters. Any suchauxiliary parameters can be set to one later.
Step 1-KP (Computing the scaling symmetry). Assuming that the evolution system forthe KP equation (12) is uniform in rank, one gets the weight-balance system,
W (u) +W (Dy) = W (v) (35)
W (u) +W (Dt) +W (Dx) = 2W (u) + 2W (Dx) = W (u) + 4W (Dx)=W (v) +W (Dy).
The parameters α and σ2 have zero weights. The solution of this linear system is
W (u) = 2W (Dx), W (v) = 4W (Dx), W (Dt) = 3W (Dx), W (Dy) = 2W (Dx). (36)
Choosing W (Dx) = 1 leads to the scaling symmetry
(x, y, t, u, v)→ (λ−1x, λ−2y, λ−3t, λ2u, λ4v). (37)
4.2. Constructing a Candidate Component
Conservation law (2) must hold on all solutions of the PDE. Therefore, the conserveddensity and its associated flux must obey the scaling symmetry of the PDE, that is, theconservation law itself must be uniform in rank. Thus, adhering to the scaling symmetry ofthe PDE, we can construct a candidate density that is a linear combination of terms of apre-selected rank.
Step 2-ZK (Building the candidate density). In this step we will construct a candidatedensity with, say, rank 6 for the ZK equation (6).
10
(a) Construct a list, P , of differential terms containing all powers of dependent variables andproducts of dependent variables that have rank 6 or less. In regard to (34), P = {u3, u2, u}.(b) Bring all of the terms in P up to rank 6 and put them into a new list, Q. This is doneby applying the total derivative operators with respect to the space variables. Taking theterms in P , u3 has rank 6 and is placed directly into Q. The term u2 has rank 4 and canbe brought up to rank six in three ways: either by applying Dx twice, or by applying Dytwice, or by applying each of Dx and Dy once. All three possibilities are considered and theresulting terms are put into Q. Similar to u2, the term u can be brought up to rank 6 infive ways, and all results are placed into Q. Doing so, P is replaced by
Q = {u3, u2x, uu2x, u
2y, uu2y, uxuy, uuxy, u4x, u3xy, u2x2y, ux3y, u4y}, (38)
in which all monomials are of rank 6.
(c) With the goal of constructing a nontrivial density with the least number of terms, removeall terms that are divergences or are divergence-equivalent to other terms in Q.
Definition 9. A term or expression f is a divergence when there exists a vector F such thatf = Div F. In the one-dimensional case, f is a total derivative when there exists a functionF such that f = DxF . Note that Dxf is essentially a one-dimensional divergence, so fromhere onwards, the term “divergence” will also cover the one-dimensional (total derivative)scenario. Two or more terms are divergence-equivalent when a linear combination of theterms is a divergence.
Removing divergences and divergence-equivalent terms is algorithmic and best accomplishedby using the Euler operator. Apply the Euler operator (19) to each term in (38) to get
Lu(x,y)Q = {3u2,−2u2x, 2u2x,−2u2y, 2u2y,−2uxy, 2uxy, 0, 0, 0, 0, 0}. (39)
By Theorem 1, divergences are terms corresponding to 0 in (39). Hence, u4x, u3xy, u2x2y,ux3y, and u4y are divergences and can be removed from Q. Now, all divergence-equivalentterms will be removed. Following Hereman et al. (2005), form a linear combination of theterms that remained in (39) with undetermined coefficients pi, and set it equal to zero,
3p1u2 − 2p2u2x + 2p3u2x − 2p4u2y + 2p5u2y − 2p6uxy + 2p7uxy = 0. (40)
After gathering like terms and forming a linear system of equations consisting of coefficients,we find that p1 = 0, p2 = p3, p4 = p5, and p6 = p7. Thus, the terms with coefficients p3, p5,and p7 are divergence-equivalent to the terms with coefficients p2, p4, and p6, respectively.For each divergence-equivalent pair, the terms with the highest order are removed from Q in(38). With all divergences and divergence-equivalent terms removed, Q = {u3, u2
x, u2y, uxuy}.
(d) A candidate density is obtained by forming a linear combination of the remaining termsin Q using undetermined coefficients ci. Thus, the candidate density with a rank of 6 forthe ZK equation is
ρ = c1u3 + c2u
2x + c3u
2y + c4uxuy. (41)
11
To determine the undetermined coefficients in (41), we must compute Dtρ and eliminate ut(and all its differential consequences). That requires that the PDE is an evolution equationor can be written in evolution form in t. The ZK equation (6) is an evolution equation.
As shown in (12), the KP equation can be written as an evolution system with respectto independent variable y. Thus, one could either interchange the variables and compute thedensity, or alternatively, construct a “candidate” for the y-component of the flux, as we willdo in a Step 2-KP below.
To complicate matters further, the conservation laws for the KP equation, (14) and(15), involve an arbitrary function f = f(t). The weight of f(t) depends on the degreeif f(t) is polynomial; whereas no weight can be assigned if f(t) is non-polynomial. Ingeneral, working with undetermined functional (instead of constant) coefficients f(x, y, t)would require a sophisticated solver for PDEs for f (see Wolf (2002)). Therefore, we cannot automatically compute (14) and (15) with our method. However, our algorithm canfind conservation laws with explicit variable coefficients, e.g., tx2, txy, etc., as long as thedegree is specified. Allowing such coefficients causes candidate densities to have a negativerank. By inspecting several conserved densities with explicit variable coefficients it is oftenpossible to conjecture (and subsequently test) the form of a conservation law with arbitraryfunctional coefficients.
Step 2-KP (Building a candidate for the y-component of the flux). For the KP equation,we will compute a candidate for the y-component of the flux with a rank equal to −3, whoseterms are multiplied by cit
nxmyp, where n+m+p ≤ 3 and n,m, and p are positive integers.
(a) Construct lists of independent variable coefficients and dependent variable terms so thatthe combined rank is equal to −3. From (37),
W (u) = 2, W (v) = 4, W (Dt) = 3, W (Dx) = 1, W (Dy) = 2. (42)
Consequently, W (t) = −3,W (x) = −1, and W (y) = −2. Table 1 shows the factors tnxmyp,up to degree 3, together with the dependent variable terms they will be multiplied to so thatthe resulting product has a rank equal to −3. Since we are computing the y-component ofthe flux, only x- and t-derivatives of dependent variables are being considered.
(b) Combine the terms in Table 1 to create a list of all possible terms with rank −3,
Q = {tx2u, xy2u, tyu, y3ux, txyux, t2ux, t
2xu2, ty2u2, t2xu2x, ty2u2x, t
2xv, ty2v
t2yuux, t2yut, t
2yu3x, t2yvx, t
3u3, t3uv, t3u2x, t
3utx, t3uu2x, t
3u4x, t3v2x}. (43)
(c) Remove all divergences and divergence-equivalent terms. Apply the Euler operator toeach term in (43). Next, linearly combine the resulting terms to get
p1
(tx2
0
)+p2
(xy2
0
)+p3
(ty0
)−p5
(ty0
)+p7
(2t2xu
0
)+p8
(2ty2u
0
)+p11
(0t2x
)+p12
(0ty2
)−p14
(2ty0
)+p17
(3t3u2
0
)+p18
(t3vt3u
)−p19
(2t3u2x
0
)+p21
(2t3u2x
0
)=0, (44)
12
Table 1: Factors tnxmyn paired with dependent variable terms whose product has a rank equal to −3.
Monomial factors tnxmyp Dependent Variable Terms
Rank Coefficient Rank Term
−5 tx2, xy2, ty 2 u−6 y3, txy, t2 3 ux−7 t2x, ty2 4 u2, u2x, v−8 t2y 5 uux, ut, u3x, vx−9 t3 6 u3, uv, u2
x, utx, uu2x, u4x, v2x
where the subscript of the undetermined coefficient, pi, corresponds to the ith term in Q.Missing pi correspond to terms that are divergences. After gathering like terms, settingtheir coefficients equal to zero, and solving the resulting linear system for the pi, we getp1 = p2 = p7 = p8 = p11 = p12 = p17 = p18 = 0, p3 = p5 + 2p14, and p19 = p21. Thus, bothterms with coefficients p5 and p14 are divergence-equivalent to the term with coefficient p3.Likewise, the term with coefficient p21 is divergence-equivalent to the term with coefficientp19. For each divergence-equivalent pair, the terms with the highest order are removed from(43). After all divergences and divergence-equivalent terms are removed
Q = {tx2u, xy2u, tyu, t2xu2, ty2u2, t2xv, ty2v, t3u3, t3uv, t3u2x}. (45)
(d) A linear combination of the terms in (45) with undetermined coefficients ci yields thecandidate (of rank −3) for the y-component of the flux,
J2 = c1tx2u+ c2xy
2u+ c3tyu+ c4t2xu2 + c5ty
2u2 + c6t2xv
+ c7ty2v + c8t
3u3 + c9t3uv + c10t
3u2x. (46)
4.3. Evaluating the Undetermined Coefficients
All, part, or none of the candidate density (41) may be an actual density for the ZKequation. It is also possible that the candidate is a linear combination of two or moreindependent densities. The true nature of the density will be revealed by computing valuesfor the undetermined coefficients. Using (2), we see that Dtρ = −Div J, so Dtρ must be adivergence with respect to the space variables. This requirement leads to an algorithm forcomputing the undetermined coefficients.
Step 3-ZK (Computing the undetermined coefficients). To compute the undeterminedcoefficients, we form a system of linear equations for these coefficients. As part of thesolution process, we also generate compatibility conditions for the constant parameters inthe PDE, if present.
(a) Compute the total t-derivative of (41),
Dtρ = 3c1u2ut + 2c2uxutx + 2c3uyuty + c4(utxuy + uxuty). (47)
13
Let E = −Dtρ after all ut, utx, etc. have been replaced using (6). This yields
E = 3c1u2(αuux + β(u3x + ux2y)) + 2c2ux(αuux + β(u3x + ux2y))x
+ 2c3uy(αuux + β(u3x + ux2y))y + c4(uy(αuux + β(u3x + ux2y))x
+ ux(αuux + β(u3x + ux2y))y). (48)
(b) By the continuity equation (2), E = Div J. Therefore, by Theorem 1, Lu(x)E ≡ 0.Apply the Euler operator to (48) and set the result identically equal to zero:
0 ≡ Lu(x,y) E = −2((3c1β + c3α)uxu2y + 2(3c1β + c3α)uyuxy
+ 2c4αuxuxy + c4αuyu2x + 3(3c1β + c2α)uxu2x
). (49)
(c) Form a linear system of equations for the undetermined coefficients by setting each factorequal to zero thus satisfying (49). After eliminating duplicate equations, the system is
3c1β + c3α = 0, c4α = 0, 3c1β + c2α = 0. (50)
(d) Check for possible compatibility conditions on the parameters α and β in (50). Thisis done by setting each ci = 1, one at a time, and algebraically eliminating the otherundetermined coefficients. Consult Goktas and Hereman (1997) for details about searchingfor compatibility conditions. System (50) is compatible for all nonzero α and β.
(e) Solve (50), taking into account the compatibility conditions (if applicable). Here,
c2 = −3βαc1, c3 = −3β
αc1, c4 = 0, (51)
where c1 is arbitrary. We set c1 = 1 so that the density (41) will be normalized on thehighest degree term, yielding
ρ = u3 − 3βα
(u2x + u2
y). (52)
Step 3-KP (Computing the undetermined coefficients).
(a) Compute the total y-derivative of (46),
DyJ2 = (c1x+ 2c4tu)txuy + c2xy(2u+ yuy) + (c3t+ 2c5tyu)(u+ yuy)
+ c6t2xvy + c7ty(2v + yvy) + 3c8t
3u2uy + c9t3(uyv + uvy) + 2c10t
3uxuxy, (53)
and replace all uy (and differential consequences) using (12). Thus,
E = −(c1x+ 2c4tu)txv − c2xy(2u+ yv)− (c3t+ 2c5tyu)(u+ yv)
+ σ2(c6t2x+ c7ty
2 + c9t3u)(utx + αu2
x + αuux + u4x)− 2c7tyv
− 3c8t3u2v − c9t3v2 − 2c10t
3uxvx. (54)
(b) Apply the Euler operator to (54) and set the result identically equal to zero. This yields
0 ≡ Lu(t,x) = −(2c2xy + (c3 − 2σ2c6)t+ 2c4t2xv + 2c5ty(2u+ yv) + 6c8t
3uv
− 2σ2c9y2(3
2ux + tutx + αtu2
x + αuu2x + tu4x)− 2c10t3v2x, c1tx
2 + c2xy2
+ (c3 + 2c7)ty + 2c4t2xu+ 2c5ty
2u+ 3c8t3u2 + 2c9t
3v − 2c10t3u2x). (55)
14
(c) Form a linear system for the undetermined coefficients ci. After duplicate equations andcommon factors have been removed, one gets
c1 = 0, c2 = 0, c3 − 2σ2c6 = 0, c3 + 2c7 = 0, c4 = 0, c5 = 0, c8 = 0, c9 = 0, c10 = 0. (56)
(d) Compute potential compatibility conditions on the parameters α and σ. Again, thesystem is compatible for all nonzero α and for σ2 = ±1.
(e) Solve the linear system, yielding
c1 = c2 = c4 = c5 = c8 = c9 = c10 = 0, c6 = 12σ2c3, c7 = −1
2c3. (57)
Set c3 = −2 (to normalize the density) and substitute the result into (46), to obtain
J2 = −2tyu− tv(σ2tx− y2). (58)
4.4. Completing the Conservation LawOnce we have one component of the conservation law, e.g., the density or a component
of the flux, we can compute the remaining components using the homotopy operator.
Step 4-ZK (Computing the flux). For (6), ρ is given in (52), so we will compute the fluxJ. Once again, by the continuity equation (2), Div J = −Dtρ = E. Therefore, we mustcompute Div−1E, where the divergence is with respect to x and y. After substitution of(51) and c1 = 1 into (48),
E = 3u2(αuux + βu3x + βux2y)− 6βαux(αuux + βu3x + βux2y)x
− 6βαuy(αuux + βu3x + βux2y)y. (59)
Now we apply the 2 D homotopy operator. The integrands (24) and (26) are
I(x)u(x,y)E = 3αu4 + β
(9u2(u2x + 2
3u2y)− 6u(3u2
x + u2y))
+ β2
α
(6u2
2x
+5u2xy + 3
2u2
2y + 32u(u2x2y + u4y)− ux(12u3x + 7ux2y)
−uy(3u3y + 8u2xy) + 52u2xu2y
), (60)
and
I(y)u(x,y)E = 3βu(uuxy − 4uxuy)− β2
2α(3u(u3xy + ux3y) + ux(13u2xy + 3u3y)
+ 5uy(u3x + 3ux2y)− 9uxy(u2x + u2y)) , (61)
respectively. The 2 D homotopy operator formulas (23) yield
H(x)u(x,y)E =
∫ 1
0
(I(x)u(x,y)E
)[λu]
dλ
λ
= 34αu4 + β
(3u2(u2x + 2
3u2y)− 2u(3u2
x + u2y))
+ β2
α
(3u2
2x + 52u2xy + 3
4u2
2y
+ 34u(u2x2y + u4y)− ux(6u3x + 7
2ux2y)− uy(3
2u3y + 4u2xy) + 5
4u2xu2y
), (62)
H(y)u(x,y)E =
∫ 1
0
(I(x)u(x,y)E
)[λu]
dλ
λ
= βu(uuxy − 4uxuy)− β2
4α(3u(u3xy + ux3y) + ux(13u2xy + 3u3y)
+ 5uy(u3x + 3ux2y)− 9uxy(u2x + u2y)) . (63)
15
The flux J =(H(x)
u(x,y)E,H(y)u(x,y)E
)has a curl term, K = (Dyθ,−Dxθ), with
θ = 2βu2uy + β2
4α
(3u(u2xy + u3y) + 5(2uxuxy + 3uyu2y + u2xuy)
). (64)
After removing K, we have conservation law (9).
Step 4-KP (Computing the density and the x-component of the flux). To complete theconservation law for (11) for which we know the y-component of the flux (58), we mustcompute the density and the x-component of the flux. By the continuity equation (2),Dtρ + DxJ1 = −DyJ2 = E. Thus, to find (ρ, J1), we must compute Div−1E, where thistime the divergence is with respect to t and x. Since we are inverting a 2 D divergence, weuse (23). After substituting (57) and c3 = −2 into (54),
E = 2tu−(t2x− σ2ty2
)(utx + αu2
x + αuu2x + u4x). (65)
The integrands for the homotopy operator are
I(t)u(t,x)E = −1
2(uDx − uxI)
∂E
∂utx= 1
2(t2u− (t2x− σ2ty2)ux), (66)
I(t)v(t,x)E = 0, (67)
I(x)u(t,x)E = u
∂E
∂ux− 1
2(uDt−utI)
∂E
∂utx−(uDx−uxI)
∂E
∂u2x
−(uD3x−uxD2
x+u2xDx−u3xI)∂E
∂u4x
= αt2u2 + t2u2x − (t2x− σ2ty2)(12ut + 2αuux + u3x) + (tx− 1
2σ2y2)u, (68)
I(x)v(t,x)E = 0. (69)
Hence,
ρ = H(t)u(t,x)E =
∫ 1
0
(I
(t)u(t,x)E + I
(t)v(t,x)E
)[λu]
dλ
λ
=
∫ 1
0
(12(t2u− (t2x− σ2ty2)ux)
)dλ = 1
2(t2u− (t2x− σ2ty2)ux), (70)
J1 = H(x)u(t,x)E =
∫ 1
0
(I
(x)u(t,x)E + I
(x)v(t,x)E
)[λu]
dλ
λ
=
∫ 1
0
(αλt2u2 + t2u2x − (t2x− σ2ty2)(1
2ut + 2αλuux + u3x) + (tx− 1
2σ2y2)u
)dλ
= 12αt2u2 + t2u2x − (t2x− σ2ty2)(1
2ut + αuux + u3x) + (tx− 1
2σ2y2)u. (71)
The solution (ρ, J1) contains a curl term, K = (Dxθ,−Dtθ), with θ = −12(t2x − σ2ty2)u.
After removing the curl term, we find(ρJ1
)=
(t2u
t2(12αu2 + u2x) + (σ2ty2 − t2x)(ut + αuux + u3x)
). (72)
The computed conservation law is the same as (14) where f = t2 and v = uy.
16
5. A Generalized Conservation Law for the KP Equation
The generalization of (72) to (14) is based on inspection of the conservation laws shownin Table 2. Indeed, these conservation laws suggest a density of the form f(t)u, where f(t) is
Table 2: Additional conservation laws for the KP equation (12).
Rank Conservation Law
5 Dt(u)
+Dx(
12αu2 + u2x − x(ut + αuux + u3x)
)−Dy
(σ2xv
)= 0
2 Dt(tu)
+Dx(t(1
2αu2 + u2x) + (1
2σ2y2 − tx)(ut + αuux + u3x)
)−Dy
(yu− v(1
2y2 − σ2tx)
)= 0.
−1 Dt(t2u)
+Dx(t2(1
2αu2 + u2x) + (σ2ty2 − t2x)(ut + αuux + u3x)
)−Dy
(2tyu− v(ty2 − σ2t2x)
)= 0
-4 Dt(t3u)
+Dx(t3(1
2αu2 + u2x) + (3
2σ2t2y2 − t3x)(ut + αuux + u3x)
)−Dy
(3t2yu− v(3
2t2y2 − σ2t3x)
)= 0
an arbitrary function. The corresponding flux would be harder to guess. However, it can becomputed as follows. Since the KP equation (12) is an evolution equation in y, we constructa suitable candidate for J2. Guided by the results in Table 2, we take
J2 = c1f′(t)yu+ c2f
′(t)y2v + c3f(t)xv, (73)
where c1, c2, and c3 are undetermined coefficients, and uy is replaced by v in agreement with(12). As before, we compute DyJ2 and replace uy and vy using (12). Doing so,
E = DyJ2 = c1f′u+ (c1 + 2c2)f
′yv − (σ2c2f′y2 + σ2c3fx)(utx + αu2
x + αuu2x + u4x). (74)
By (2), DyJ2 = −Div(ρ, J1). Therefore, by Theorem 1,
0 ≡ Lu(t,x)E =
((c1 − σ2c3)f
′
(c1 + 2c2)f′y
). (75)
Clearly, c2 = −12c1 and c3 = σ2c1. If we set c1 = −1, we obtain the y-component of (14),
where v = uy. Application of the homotopy operator (in this case to an expression witharbitrary functional coefficients) yields (ρ, J1).
Due to the presence of an arbitrary function f(t), it was impossible to directly compute(14) with our algorithm. However, pattern matching with the results in Table 2 and some in-teractive work lead to (14), which can be then be verified with ConservationLawsMD.m.Conservation law (15) was obtained in a similar way.
17
6. Applications
In this section we apply our algorithm to a variety of (2+1)- and (3+1)-dimensional non-linear PDEs. These PDEs highlight many of the issues that arise when using the algorithm.
6.1. The Sawada-Kotera Equation in 2 D
The (1+1)-dimensional Sawada-Kotera (SK) equation belongs to a family of fifth-orderKdV-type equations. As shown by Goktas and Hereman (1997), the SK equation has in-finitely many conservation laws and is known to be completely integrable. The (2+1)-dimensional SK equation,
ut = 5u2ux + 5uu3x + 5uuy + 5uxu2x + 5u2xy + u5x − 5∂−1x u2y + 5ux∂
−1x uy, (76)
with u(x, y, t) was derived by Konopelchenko and Dubrovsky (1984) as a higher-dimensionalcompletely integrable equation. Our algorithm can not handle the integral terms like ∂−1
x f =∫fdx in (76). Therefore, setting v = ∂−1
x uy allows us to write (76) as a system of evolutionequations in y :
vy = −1
5ut + u2ux + uu3x + uvx + uxu2x + v3x +
1
5u5x + uxv, uy = vx. (77)
Application of our algorithm to (77) then yields
Dt(fu)−Dx
(5f(1
3u3 + uv + uu2x + uxy + 1
5u4x)− f ′yv
)+Dy
(5fv − f ′yu
)= 0, (78)
Dt(fyu
)−Dx
(5fy(1
3u3 + uv + uu2x + uxy + 1
5u4x) + (5fx− 1
2f ′y2)v
)+Dy
((5fx− 1
2f ′y2)u+ 5fyv
)= 0, (79)
where f = f(t) is an arbitrary function.Note that the densities in (78) and (79) are identical to those in (14) and (15) for the
KP equation. These two densities occur often in (2+1)-dimensional PDEs that have a utxinstead of a ut term, as show in the next example.
6.2. The Khokhlov-Zabolotskaya Equation in 2 D and 3 D
The Khokhlov-Zabolotskaya (KZ) equation or dispersionless KP equation, describes thepropagation of sound in non-linear media in two or three space dimensions (Sanders andWang, 1997a). The (2+1)-dimensional KZ equation,
(ut − uux)x − u2y = 0, (80)
with u(x, y, t) can be written as a system of evolution equations in y,
uy = v, vy = utx − u2x − uu2x, (81)
18
by setting v = uy. Again, two familiar densities appear in the following conservation laws,
Dt(ux)−Dx(uux)−Dy(uy) = 0, (82)
Dt(fu)−Dx
(12fu2 + (fx+ 1
2f ′y2)(ut − uux)
)+Dy
((fx+ 1
2f ′y2)uy−f ′yu
)= 0, (83)
Dt(fyu
)−Dx
(12fyu2+(fxy + 1
6f ′y3)(ut − uux)
)+Dy
((fxy + 1
6f ′y3)uy−(fx+ 1
2f ′y2)u
)= 0. (84)
The computation of conservation laws for the (3+1)-dimensional KZ equation,
(ut − uux)x − u2y − u2z = 0, (85)
where u(x, y, z, t), is more difficult. This equation can be written as a system of evolutionequations in either y or z. Although the intermediate results will differ, either choice willlead to equivalent conservation laws. Writing (85) as an evolution system in z,
uz = v, vz = utx − u2x − uu2x − u2y, (86)
we were able to compute a variety of conservation laws whose densities are shown in Table 3.
Table 3: Densities for the (3+1)-dimensional KZ equation (85).
Rank Densities
2 ρ1 = xux0 ρ2 = xyux, ρ3 = xzux−1 ρ4 = tu
−2 ρ5 = xyzux, ρ6 = (y2 − z2)xux−3 ρ7 = tyu, ρ8 = tzu
−4 ρ9 = t2u, ρ10 = (y3 − 3yz2)xux, ρ11 = (3yz2 − z3)xux−5 ρ12 = tyzu, ρ13 = (y2 − z2)tu
−6 ρ14 = t2yu, ρ15 = t2xzu, ρ16 = (y3z − yz3)xux,
ρ17 = (y4 − 6y2z2 + z4)xux−7 ρ18 = t3u, ρ19 = (y3 − 3yz2)tu, ρ20 = (3y2z − z3)tu
−8 ρ21 = t2yzu, ρ22 = (y2 − z2)t2u, ρ23 = (y5 − 10y3z2 + 5yz4)xux,
ρ24 = (5y4z − 10y2z3 + z5)xux
Density ρ1 = xux in Table 3 is part of conservation law
Dt(xux
)+Dx
(12u2 − xuux
)−Dy
(xuy
)−Dz
(xuz
)= 0, (87)
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which can be rewritten as
Dt(u)−Dx
(12u2 + x(ut − uux)
)+Dy
(xuy
)+Dz
(xuz
)= 0 (88)
by swapping terms in the density and the x-component of the flux. More general, if a factorxux appears in a density then that factor can be replaced by u. Doing so, all densitiesin Table 3 that can be expressed as ρ = P (y, z, t)u. Introducing arbitrary functions f =f(y, z, t) and g = g(y, z, t), the conservation laws corresponding to the densities in Table 3can be summarized as
Dt(fu)−Dx
(12fu2 + (fx+ g)(ut − uux)
)−Dy
((fyx+ gy)u− (fx+ g)uy
)−Dz
((fzx+ gz)u− (fx+ g)uz
)= −(f2y + f2z)xu− (g2y + g2z − ft)u. (89)
Equation (89) is only a conservation law when the constraints ∆f = 0 and ∆g = ft aresatisfied, where ∆ = ∂2
∂y2+ ∂2
∂z2. Thus f must be a harmonic function and g must satisfy the
Poisson equation with ft on the right hand side. Combining both equations produces thebiharmonic equation ∆(∆g) = 0. As shown by Tikhonov and Samarskii (1963), ∆2g = 0has general solutions of the form
g = y g1(y, z) + g2(y, z) and g = z g1(y, z) + g2(y, z), (90)
where ∆g1 = 0 and ∆g2 = 0. Treating t as a parameter, four solutions for g(y, z, t) are
g(y, z, t) = 12y ∂−1
y ft(y, z, t), (91)
g(y, z, t) = 12∂−1y (yft) = 1
2(y ∂−1
y ft(y, z, t)− ∂−2y ft(y, z, t)), (92)
g(y, z, t) = 12z ∂−1
z ft(y, z, t), (93)
g(y, z, t) = 12∂−1z (zft) = 1
2(z ∂−1
z ft(y, z, t)− ∂−2z ft(y, z, t)). (94)
Thus, g can be written in terms of f. For every conservation law corresponding to thedensities in Table 3, g can be computed explicitly using one of the equations in (91)-(94).
Conservation laws for the KZ equation have been reported in literature by Sharomet(1989) and Sanders and Wang (1997a). However, substitution of these results into (2) revealsinaccuracies. After bringing the mistake to their attention, Sanders and Wang (1997b) havesince corrected their conservation law.
6.3. The Camassa-Holm Equation in 2 D
The (2+1)-dimensional Camassa-Holm (CH) equation,
(ut + κux − ut2x + 3uux − 2uxu2x − uu3x)x + u2y = 0, (95)
for u(x, y, t) was derived in a study of water waves (Johnson, 2002) as an extension of the(1+1)-dimensional CH equation, which is a completely integrable PDE due to Camassaand Holm (1993). A study by Gordoa et al. (2004), concluded that (95) is not completelyintegrable.
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The CH equation (95) can be written as a system of evolution equations in y. Indeed,
uy = v, vy = −(αut + κux − ut2x + 3βuux − 2uxu2x − uu3x)x. (96)
Note that we introduced auxiliary parameters α and β as coefficients of the ut and uuxterms, respectively. Analysis of the weight-balance equations in Poole (2009) shows thatα, β and κ must carry weights to make (95) and (96) scaling invariant.
Without conditions on these parameters, we found two conservation laws:
Dt(fu)
+Dx(
1αf(3
2βu2+ κu− 1
2u2x− uu2x − utx)− ( 1
αfx− 1
2f ′y2)(αut+ κux
+ 3βuux − 2uxu2x − uu3x − ut2x))−Dy
(( 1αfx− 1
2f ′y2)uy + f ′yu
)= 0, (97)
Dt(fyu
)+Dx
(1αfy(3
2βu2+κu− 1
2u2x−uu2x−utx)−y( 1
αfx− 1
6f ′y2)(αut+κux
+3βuux−2uxu2x−uu3x−ut2x))−Dy
(( 1αfx− 1
6f ′y2)yuy−( 1
αfx− 1
2f ′y2)u
)= 0, (98)
where f(t) is an arbitrary function. By setting α = β = 1, the conservation laws correspondto (95). When adding weighted parameters to an equation, it is possible that the programreturns conservation laws accompanied by conditions on the weighted parameters. If suchconditions are not satisfied for α = β = 1, then the conservation laws should be discarded.
7. Using the Program ConservationLawsMD.m
Before using ConservationLawsMD.m, all data files provided with the program, aswell as additional data files created by the user, must be placed into one directory. Next,open the Mathematica notebook ConservationLawsMD.nb which contains instructionsfor loading the code. Executing the command ConservationLawsMD[] will open a menu,offering the choice of computing conservation laws for a PDE from the menu or from a datafile prepared by the user. All PDEs listed in the menu have matching data files. Anexample of a data file is shown in Figure 1. The independent space variables must be x,y, and z. The symbol t must be used for time. Dependent variables must be entered as ui,i = 1, . . . , N, where N is the total number of dependent variables. In a (1+1)-dimensionalcase, the dependent variables (in Mathematica syntax) are u[1][x,t], u[2][x,t], etc. Ina (3+1)-dimensional cases, u[1][x,y,z,t], etc., where t is always the last argument.
8. Conclusions
We have presented an algorithm and a software package, ConservationLawsMD.m, tocompute conservation laws of nonlinear polynomial PDEs in multiple space dimensions. Thealgorithm uses tools from calculus, the calculus of variations, linear algebra, and differentialgeometry. Conservation laws have been computed for a variety of multi-dimensional PDEs,demonstrating the versatility of the software package.
The software is easy to use, runs fast, and has been tested on a variety of PDEs. Many ofthe test cases have been added to the menu of the program. In addition, the program allows
21
(* data file d kd2d.m *)
(* Menu item 2-10 *)
(*** (2+1)-dimensional Gardner equation ***)
(* as proposed by Konopelchenko and Dubrovsky (1984) *)
eq[1] = D[u[1][x,y,t],y] - D[u[2][x,y,t],x];
eq[2] = - D[u[1][x,y,t],t] + D[u[1][x,y,t],x,3]
+ 6*beta*u[1][x,y,t]*D[u[1][x,y,t],x]
- (3/2)*alpha∧2*u[1][x,y,t]∧2*D[u[1][x,y,t],x]
+ 3*D[u[2][x,y,t],y] - 3*alpha*D[u[1][x,y,t],x]*u[2][x,y,t];
diffFunctionListINPUT = {eq[1],eq[2]};numDependentVariablesINPUT = 2;
independentVariableListINPUT = {x,y};nameINPUT = "(2+1)-dimensional Gardner Equation";
noteINPUT = "Any additional information can be put here.";
parametersINPUT = {alpha};All parameters without weight must be placed in this list.
weightedParametersINPUT = {beta};Parameters that carry a weight must be placed in this list.
userWeightRulesINPUT = {};Optional: the user can choose weights for variables.
rankRhoINPUT = Null;Can be changed to a list of values if the user wishes to test several ranks at onetime. The program runs automatically when such values are given.
explicitIndependentVariablesInDensitiesINPUT = Null;Can be set to 0, 1, 2, . . . , specifying the maximum degree (n+m+p) of coefficientscit
nxmyp in the density.formRhoINPUT = {};
The user can give a density to be tested. This works only for evolution equationsin variable t.
(* end of data file d kd2d.m *)
Figure 1: Data file for the (2+1)-dimensional Gardner equation.
the user to test conservation laws either computed with other methods, obtained from theliterature, or conjectured after work with the code. The latter is particularly relevant forfinding conservation laws involving arbitrary functions, cf. the KP equation.
Currently, ConservationLawsMD.m has two major limitations: (i) the PDE musteither be an evolution equation or must correspond to a system of evolution equations, per-haps after an interchange of variables or some other transformation; and (ii) the programcan only generate local polynomial densities and fluxes. The testing capabilities of Con-servationLawsMD.m are quite versatile. The code can be used to test conservation laws
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involving smooth functions of the independent variables and the densities and fluxes are notrestricted to polynomial differential functions.
Future versions of the program will be able to handle PDEs with mixed derivatives,transcendental nonlinearities, and PDEs that are not of evolution type.
Acknowledgements
This material is based in part upon work supported by the National Science Foundation(NSF) under Grant No. CCF-0830783. Any opinions, findings, and conclusions or recom-mendations expressed in this material are those of the authors and do not necessarily reflectthe views of NSF.
Mark Hickman (University of Canterbury, Christchurch, New Zealand) and BernardDeconinck (University of Washington, Seattle) are gratefully acknowledged for valuable dis-cussions. Undergraduate students Jacob Rezac, John-Bosco Tran, and Travis “Alan” Volzare thanked for their help with this project.
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