Conservation Laws of Some Physical Models via Symbolic Package GeM

Abstract

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation. The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliers and conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and first homotopy and second homotopy formulas.

1. Introduction

The study of conservation laws plays a vital role in analysis, solution, and reductions of PDEs. For the PDEs, the conservation laws are used in wide variety of applications, for example, inverse scattering transform in soliton solutions [1], bi-Hamiltonian structures and recursion operators [2], Lax operators [3], and derivation of conserved quantities for jet flows [4].

Different methods have been developed so far for the construction of conservation laws and are well documented in [5–7]. In the last few decades, the researchers focused on the development of symbolic computational packages based on different approaches of conservation laws. These packages work with either Mathematica or Maple. The development of symbolic computational packages gives relief to perform complicated and tedious algebraic computation. Recently, several computational packages have been developed, for example, CONDENS.M by Göktaş and Hereman [8], RUDCE by Wolf et al. [9–11], TransPDEDensity.m by Adams and Hereman [12], GeM by Cheviakov [13, 14], Vessiot suite by Anderson and Cheb-Terrab [15], ConservationLawsMD.m by Poole and Hereman [16], and SADE by Rocha Filho and Figueiredo [17].

In this paper, we will use GeM package [13] to compute the conservation laws for partial differential equations (PDEs) arising in applications. GeM package works with Maple to obtain the symmetries and conservation laws of differential equations. In symmetry analysis, it first computes the overdetermined system of determining equations and then simplifies the system by Rif package routines. After simplification, a Maple command in GeM generates all symmetry generators of differential equation. In conservation laws analysis, GeM computes an overdetermined system of determining equation of conservation law multipliers, and then this system is simplified by Rif package which is solved by using the built-in Maple function pdsolve to get multipliers. After computing multipliers, the conservation laws fluxes are derived by one of the following four methods: direct method [18, 19], first homotopy formula [20], second homotopy formula [19], and scaling symmetry formula [21]. All these four methods have some limitations in their use. The direct method written in GeM [13] is a Maple implementation based on Wolf [11] program in REDUCE. For simple partial differential equation (PDE) systems and multipliers, direct method is used to calculate fluxes. It is also used if arbitrary functions are involved. The conservation laws fluxes for complicated PDEs or multipliers, not involving arbitrary functions, are established by using first and second homotopy formulas. The scaling symmetry method is used to compute fluxes for the scaling-homogeneous PDEs or/and multipliers. For the complicated scaling-homogeneous PDEs and/or multipliers involving arbitrary functions, this is only a systematic method for computing fluxes.

The evolution equations are important and arise in many applications. We compute the conservation laws of various nonlinear evolution equations using GeM Maple routines. This includes a (1+1)-dimensional evolution equation [22], lubrication models [23], sinh-Poisson equation [24], Kaup-Kupershmidt equation [25], and modified Sawada-Kotera equation [26]. At last, we summarize and discuss our results.

2. Multipliers and Conservation Laws Using GeM Maple Routines

2.1. Evolution Equation

As a first example, consider the following evolution equation [22]: 𝑢_𝑡𝑡+𝑎𝑢_𝑥𝑥+𝑏𝑢+𝑐𝑢³=0, (1) where 𝑢(𝑡,𝑥) and 𝑎,  𝑏,  𝑐 are constants. We will explain this example in detail along with GeM Maple routines given in [13, 14]. The variables and partial differential equation (PDE) (1) are defined in GeM by the following Maple commands. With(GeM): gem_decl_vars(indeps=[t,x], deps=[u(t,x)]); gem_decl_eqs([diff(u(t,x),t,t)+a*diff(u(t,x),x,x)+b*u(t,x)+c*u³(t,x)=0], solve_for=[diff(u(t,x),t,t)]).The option solve_for is used in the flux-computation routine, and actually it defines a set of leading derivatives the given PDE systems can be solved for.

Consider multipliers of the form Λ=Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥). In GeM, we use the Maple routines, det_eqs:=gem_conslaw_det_eqs([t,x,u(t,x),diff(u(t,x),t),diff(u(t,x),x)]): CL_multipliers:=gem_conslaw_multipliers(); simplified_eqs:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1),to obtain the set of determining equations for the multipliers expressed in the simplified form as Λ_𝑥𝑥=0,Λ_𝑢=0,Λ_𝑥𝑢𝑥=0,Λ_{𝑢𝑥𝑢𝑥}Λ=0,_𝑡=−𝑎𝑢_𝑥Λ_𝑥𝑢_𝑡,Λ_𝑢𝑡=Λ−𝑢_𝑥Λ_𝑢𝑥𝑢_𝑡,with𝑎≠0,𝑏≠0,𝑐≠0. (2)

To solve the system (2), we use the Maple command multipliers_sol:=pdsolve(simplified_eqs[Solved]),and it yields Λ𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥=𝑐₃𝑥+𝑐₁𝑢_𝑡+−𝑐₃𝑎𝑡+𝑐₂𝑢_𝑥, (3) where 𝑐₁,  𝑐₂,  𝑐₃ are arbitrary constants. We obtain three linearly independent conservation laws, arising from the multipliers Λ⁽¹⁾=𝑢_𝑡,Λ⁽²⁾=𝑢_𝑥,Λ⁽³⁾=𝑥𝑢_𝑡−𝑎𝑡𝑢_𝑥. (4) Next step is the derivation of conservation laws associated with multipliers given in (4). The Maple command gem_get_CL_fluxes(multipliers_sol)computes the flux expressions by the direct method. For the multipliers (4), we have the following conservation laws fluxes: 𝜙⁽¹⁾=12𝑢²_𝑡−12𝑎𝑢²_𝑥+14𝑐𝑢⁴+12𝑏𝑢²,𝜓⁽¹⁾=𝑎𝑢_𝑥𝑢_𝑡,𝜙⁽²⁾=𝑢_𝑡𝑢_𝑥+𝑏𝑡𝑢𝑢_𝑥+𝑐𝑡𝑢³𝑢_𝑥,𝜓⁽²⁾1=−2𝑢²_𝑡−𝑐𝑡𝑢³𝑢_𝑡−𝑏𝑡𝑢𝑢_𝑡+12𝑎𝑢²_𝑥,𝜙⁽³⁾1=−2𝑎𝑥𝑢²_𝑥−𝑎𝑡𝑢_𝑡𝑢_𝑥+12𝑥𝑢²_𝑡+14𝑐𝑥𝑢⁴+12𝑏𝑥𝑢²,𝜓⁽³⁾1=−4𝑎𝑐𝑡𝑢⁴−12𝑎𝑏𝑡𝑢²+12𝑎𝑡𝑢²_𝑡+𝑎𝑥𝑢_𝑥𝑢_𝑡−12𝑎²𝑡𝑢²_𝑥. (5) The multipliers given in (4) do not involve arbitrary functions, so homotopy formulas can be used to compute fluxes. We call the routine for first homotopy method  gem_get_CL_fluxes(multipliers_sol, method=“Homotopy1”)to get the following expressions for conservation law fluxes: 𝜙⁽¹⁾=14𝑐𝑢⁴+12𝑢²_𝑡+12𝑎𝑢𝑢_𝑥𝑥+12𝑏𝑢²,𝜓⁽¹⁾1=−2𝑎𝑢𝑢_𝑡𝑥+12𝑎𝑢_𝑥𝑢_𝑡,𝜙⁽²⁾1=−2𝑢𝑢_𝑡𝑥+12𝑢_𝑡𝑢_𝑥,𝜓⁽²⁾=14𝑐𝑢⁴+12𝑎𝑢²_𝑥+12𝑢𝑢_𝑡𝑡+12𝑏𝑢²,𝜙⁽³⁾=14𝑐𝑢⁴1𝑥+2𝑎𝑢𝑢_𝑥+12𝑎𝑡𝑢𝑢_𝑡𝑥−12𝑎𝑡𝑢_𝑡𝑢_𝑥+12𝑥𝑢²_𝑡+12𝑎𝑥𝑢𝑢_𝑥𝑥+12𝑏𝑥𝑢²,𝜓⁽³⁾1=−4𝑎𝑐𝑡𝑢⁴−12𝑎𝑢𝑢_𝑡−12𝑎𝑥𝑢𝑢_𝑡𝑥−12𝑎²𝑡𝑢²_𝑥+12𝑎𝑥𝑢_𝑡𝑢_𝑥−12𝑎𝑡𝑢𝑢_𝑡𝑡−12𝑎𝑏𝑡𝑢². (6) For second homotopy formula, the Maple command  gem_get_CL_fluxes(multipliers_sol, method=“Homotopy2”)yields divergence expressions in the same form as in (6).

The PDE (1) has no scaling symmetry; therefore, we cannot apply the scaling symmetry formula here for derivation of fluxes.

2.2. Lubrication Models

Now we will study two lubrication models for conservation laws point of view. Gandarias and Medina [23] performed the symmetry analysis of lubrication model 𝑢_𝑡=𝑓(𝑢)𝑢_{𝑥𝑥𝑥𝑥}, (7) where 𝑓 is an arbitrary function. For 𝑓(𝑢)=𝑐(𝑢+𝑏)^𝑎 and 𝑓(𝑢)=𝛾𝑒^𝛼𝑢, this equation has some extra symmetry [23]. Without loss of generality, take 𝑓(𝑢)=𝑢+𝑏 in (7); we have 𝑢_𝑡=(𝑢+𝑏)𝑢_{𝑥𝑥𝑥𝑥}, (8) where 𝑏 is arbitrary constant. Consider the multipliers of form Λ(𝑡,𝑥,𝑢) in GeM Maple routines, and then we obtain the following four multipliers: Λ⁽¹⁾()=1𝑡,𝑥,𝑢(𝑢+𝑏),Λ⁽²⁾()=𝑥𝑡,𝑥,𝑢(),Λ𝑢+𝑏₍₃₎()=𝑥𝑡,𝑥,𝑢²2(𝑢+𝑏),Λ⁽⁴⁾()=𝑥𝑡,𝑥,𝑢³6().𝑢+𝑏 (9) The fluxes associated with the multipliers given in (7) are computed by homotopy first method and are given by 𝜙⁽¹⁾=ln𝑢+𝑏𝑏,𝜓⁽¹⁾=𝑢_𝑥𝑥𝑥,𝜙⁽²⁾=𝑥ln𝑢+𝑏𝑏,𝜓⁽²⁾=−𝑢_𝑥𝑥+𝑥𝑢_𝑥𝑥𝑥,𝜙⁽³⁾=12𝑥²ln𝑢+𝑏𝑏,𝜓⁽³⁾=𝑢_𝑥−𝑥𝑢_𝑥𝑥+𝑥²𝑢_𝑥𝑥𝑥2,𝜙⁽⁴⁾=16𝑥³ln𝑢+𝑏𝑏,𝜓⁽⁴⁾=−𝑢+𝑥𝑢_𝑥−𝑥²𝑢_𝑥𝑥2+𝑥³𝑢_𝑥𝑥𝑥6. (10) We will get the same fluxes for (7) if we define higher order multipliers in GeM Maple routines.

Another interesting lubrication model is 𝑢_𝑡+𝑢_{𝑥𝑥𝑥𝑥}𝑒^𝑢=0. (11) It is obtained by taking 𝑓(𝑢)=𝑒^−𝑢 in (7). The GeM Maple routines yield the following four multipliers of form Λ(𝑡,𝑥,𝑢): Λ⁽¹⁾(𝑡,𝑥,𝑢)=𝑒^𝑢,Λ⁽²⁾(𝑡,𝑥,𝑢)=𝑥𝑒^𝑢,Λ⁽³⁾()=1𝑡,𝑥,𝑢2𝑥²𝑒^𝑢,Λ⁽⁴⁾()=1𝑡,𝑥,𝑢6𝑥³𝑒^𝑢. (12)

The corresponding fluxes obtained by homotopy first method are 𝜙⁽¹⁾=−1+𝑒^𝑢,𝜓⁽¹⁾=𝑢_𝑥𝑥𝑥,𝜙⁽²⁾=𝑥−1+𝑒^𝑢,𝜓⁽²⁾=−𝑢_𝑥𝑥+𝑥𝑢_𝑥𝑥𝑥,𝜙⁽³⁾=12𝑥²−1+𝑒²,𝜓⁽³⁾=𝑢_𝑥−𝑥𝑢_𝑥𝑥+𝑥²𝑢_𝑥𝑥𝑥2,𝜙⁽⁴⁾=16𝑥³−1+𝑒^𝑢,𝜓⁽⁴⁾=−𝑢+𝑥𝑢_𝑥−𝑥²𝑢_𝑥𝑥2+𝑥³𝑢_𝑥𝑥𝑥6. (13) The conservation laws fluxes derived here can be used to find the solution of lubrication models and will be considered in future work.

2.3. sinh-Poisson Equation

The (2+1)-dimensional sinh-Poisson equation is [24] 𝑢_𝑥𝑥+𝑢_𝑧𝑧−𝑢_𝑡𝑡=𝜆²sinh𝑢, (14) where 𝑢(𝑡,𝑥,𝑧). The conservation laws for PDE (14) are derived here by using GeM routines. Consider the multipliers of form Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑧) in GeM routines, then it will yield six multipliers not containing any arbitrary function. The expression for fluxes is computed by using first homotopy formula. The multipliers and associated conserved vectors computed by first homotopy formula are given in Table 1.

Table 1 

Multipliers and conserved vectors for PDE (14).

2.4. Kaup-Kupershmidt Equation

Now, we will compute the conservation laws for the fifth order Kaup-Kupershmidt [25]: 𝑢_𝑡=𝑢_{𝑥𝑥𝑥𝑥𝑥}+5𝑢𝑢_𝑥𝑥𝑥+252𝑢_𝑥𝑢_𝑥𝑥+5𝑢²𝑢_𝑥. (15) The GeM Maple routines yield three multipliers of the form Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥) for PDE (14). The first homotopy formula is applied to derive the expressions for conservation laws fluxes. One more multiplier can be computed if we consider higher order multipliers of the form Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_𝑥𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥}). All the multipliers and associated conserved vectors for PDE (14) computed by first homotopy formula are presented in Table 2.

Table 2 

Multipliers and conserved vectors for PDE (15).

2.5. Modified Sawada-Kotera Equation

Consider the fifth order modified SK equation: 𝑢_𝑡=𝑢_{𝑥𝑥𝑥𝑥𝑥}−5𝑢_𝑥𝑢_𝑥𝑥+5𝑢𝑢²_𝑥+5𝑢²𝑢_𝑥𝑥−𝑢⁵_𝑥. (16) For PDE (16), two conserved densities were derived by first computing Lax pair (see [26]). The higher order conservation laws fluxes exist for higher order multipliers and are not reported in [26]. Consider the multipliers of form Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑡𝑥,𝑢_𝑥𝑥,𝑢_𝑥𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥}) in GeM routines, then it will yield two simple and three higher order multipliers not containing any arbitrary function. The simple multipliers yield same fluxes as derived in [26], and three new fluxes corresponding to higher order multipliers are computed. The multipliers and associated conserved vectors computed by first homotopy formula are listed in Table 3.

Table 3 

Multipliers and conserved vectors for PDE (16).

3. Conclusions

The conservation laws for the evolution equation, Benjamin equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation were derived by using the symbolic software GeM. First of all, we considered the evolution equation, and the commands for all GeM Maple routines, were explicitly given. The first order multipliers were defined in GeM Maple routines and three multipliers were obtained. The expressions for fluxes were computed by direct method and first and second homotopy formulas and equivalent expressions for fluxes were obtained. The scaling symmetry method was not applicable here as no scaling symmetry exists for the nonlinear evolution equation. The conservation laws fluxes for the lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation were derived by the first homotopy formula. For the modified Sawada-Kotera equation, three new fluxes were derived.

The fluxes derived here can be used in constructing the solutions of underlying PDEs and will be considered in the future work.

Conflict of Interests

The authors declare that there is no conflict of interests.