Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 897912, 7 pageshttp://dx.doi.org/10.1155/2013/897912

Research ArticleConservation Laws of Some Physical Models viaSymbolic Package GeM

Rehana Naz,1 Imran Naeem,2 and M. Danish Khan2

1 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan2Department of Mathematics, School of Science and Engineering, LUMS, Lahore Cantt 54792, Pakistan

Correspondence should be addressed to Rehana Naz; rehananaz qau@yahoo.com

Received 10 May 2013; Accepted 13 June 2013

Academic Editor: Chaudry Masood Khalique

Copyright © 2013 Rehana Naz et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation,andmodified Sawada-Kotera equation.The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliersand conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and firsthomotopy 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 con-servation laws are used in wide variety of applications, forexample, inverse scattering transform in soliton solutions [1],bi-Hamiltonian structures and recursion operators [2], Laxoperators [3], and derivation of conserved quantities for jetflows [4].

Different methods have been developed so far for theconstruction of conservation laws and are well documentedin [5–7]. In the last few decades, the researchers focusedon the development of symbolic computational packagesbased on different approaches of conservation laws. Thesepackages work with either Mathematica or Maple. Thedevelopment of symbolic computational packages gives reliefto perform complicated and tedious algebraic computation.Recently, several computational packages have been devel-oped, for example, CONDENS.M by Goktas and Hereman[8], RUDCE by Wolf et al. [9–11], TransPDEDensity.m byAdams and Hereman [12], GeM by Cheviakov [13, 14],Vessiot suite by Anderson and Cheb-Terrab [15], Conserva-tionLawsMD.m by Poole and Hereman [16], and SADE byRocha Filho and Figueiredo [17].

In this paper, we will use GeM package [13] to com-pute the conservation laws for partial differential equations

(PDEs) arising in applications. GeM package works withMaple to obtain the symmetries and conservation laws ofdifferential equations. In symmetry analysis, it first computesthe overdetermined system of determining equations andthen simplifies the system by Rif package routines. Aftersimplification, a Maple command in GeM generates allsymmetry generators of differential equation. In conservationlaws analysis, GeM computes an overdetermined systemof determining equation of conservation law multipliers,and then this system is simplified by Rif package which issolved by using the built-in Maple function pdsolve to getmultipliers. After computing multipliers, the conservationlaws fluxes are derived by one of the following four methods:direct method [18, 19], first homotopy formula [20], secondhomotopy 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 implemen-tation based on Wolf [11] program in REDUCE. For simplepartial differential equation (PDE) systems and multipliers,direct method is used to calculate fluxes. It is also usedif arbitrary functions are involved. The conservation lawsfluxes for complicated PDEs or multipliers, not involvingarbitrary functions, are established by using first and secondhomotopy formulas. The scaling symmetry method is usedto compute fluxes for the scaling-homogeneous PDEs or/andmultipliers. For the complicated scaling-homogeneous PDEs

2 Mathematical Problems in Engineering

and/or multipliers involving arbitrary functions, this is onlya systematic method for computing fluxes.

The evolution equations are important and arise in manyapplications. We compute the conservation laws of variousnonlinear 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-Koteraequation [26]. At last, we summarize and discuss our results.

2. Multipliers and Conservation LawsUsing GeM Maple Routines

2.1. Evolution Equation. As a first example, consider thefollowing evolution equation [22]:

𝑢𝑡𝑡+ 𝑎𝑢𝑥𝑥+ 𝑏𝑢 + 𝑐𝑢

3= 0, (1)

where 𝑢(𝑡, 𝑥) and 𝑎, 𝑏, 𝑐 are constants. We will explain thisexample 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 givenPDE systems can be solved for.

Consider multipliers of the form Λ = Λ(𝑡, 𝑥, 𝑢, 𝑢𝑡, 𝑢𝑥). 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 multipliersexpressed in the simplified form as

Λ𝑥𝑥= 0, Λ

𝑢= 0, Λ

𝑥𝑢𝑥

= 0, Λ𝑢𝑥𝑢𝑥

= 0,

Λ𝑡= −𝑎𝑢𝑥Λ𝑥

𝑢𝑡

,

Λ𝑢𝑡

=Λ − 𝑢

𝑥Λ𝑢𝑥

𝑢𝑡

, with 𝑎 = 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

Λ (𝑡, 𝑥, 𝑢, 𝑢𝑡, 𝑢𝑥) = (𝑐3𝑥 + 𝑐1) 𝑢𝑡+ (−𝑐3𝑎𝑡 + 𝑐2) 𝑢𝑥, (3)

where 𝑐1, 𝑐2, 𝑐3are arbitrary constants. We obtain three

linearly independent conservation laws, arising from themultipliers

Λ(1)= 𝑢𝑡, Λ

(2)= 𝑢𝑥, Λ

(3)= 𝑥𝑢𝑡− 𝑎𝑡𝑢𝑥. (4)

Next step is the derivation of conservation laws associatedwith multipliers given in (4). The Maple command

gem get CL fluxes(multipliers sol)computes the flux expressions by the direct method. Forthe multipliers (4), we have the following conservation lawsfluxes:

𝜙(1)=1

2𝑢2

𝑡−1

2𝑎𝑢2

𝑥+1

4𝑐𝑢4+1

2𝑏𝑢2, 𝜓

(1)= 𝑎𝑢𝑥𝑢𝑡,

𝜙(2)= 𝑢𝑡𝑢𝑥+ 𝑏𝑡𝑢𝑢

𝑥+ 𝑐𝑡𝑢3𝑢𝑥,

𝜓(2)= −1

2𝑢2

𝑡− 𝑐𝑡𝑢3𝑢𝑡− 𝑏𝑡𝑢𝑢

𝑡+1

2𝑎𝑢2

𝑥,

𝜙(3)= −1

2𝑎𝑥𝑢2

𝑥− 𝑎𝑡𝑢𝑡𝑢𝑥+1

2𝑥𝑢2

𝑡+1

4𝑐𝑥𝑢4+1

2𝑏𝑥𝑢2,

𝜓(3)= −1

4𝑎𝑐𝑡𝑢4−1

2𝑎𝑏𝑡𝑢2+1

2𝑎𝑡𝑢2

𝑡+ 𝑎𝑥𝑢

𝑥𝑢𝑡−1

2𝑎2𝑡𝑢2

𝑥.

(5)

The multipliers given in (4) do not involve arbitrary func-tions, 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)=1

4𝑐𝑢4+1

2𝑢2

𝑡+1

2𝑎𝑢𝑢𝑥𝑥+1

2𝑏𝑢2,

𝜓(1)= −1

2𝑎𝑢𝑢𝑡𝑥+1

2𝑎𝑢𝑥𝑢𝑡,

𝜙(2)= −1

2𝑢𝑢𝑡𝑥+1

2𝑢𝑡𝑢𝑥,

𝜓(2)=1

4𝑐𝑢4+1

2𝑎𝑢2

𝑥+1

2𝑢𝑢𝑡𝑡+1

2𝑏𝑢2,

𝜙(3)=1

4𝑐𝑢4𝑥 +1

2𝑎𝑢𝑢𝑥+1

2𝑎𝑡𝑢𝑢𝑡𝑥−1

2𝑎𝑡𝑢𝑡𝑢𝑥

+1

2𝑥𝑢2

𝑡+1

2𝑎𝑥𝑢𝑢𝑥𝑥+1

2𝑏𝑥𝑢2,

𝜓(3)= −1

4𝑎𝑐𝑡𝑢4−1

2𝑎𝑢𝑢𝑡−1

2𝑎𝑥𝑢𝑢𝑡𝑥−1

2𝑎2𝑡𝑢2

𝑥

+1

2𝑎𝑥𝑢𝑡𝑢𝑥−1

2𝑎𝑡𝑢𝑢𝑡𝑡−1

2𝑎𝑏𝑡𝑢2.

(6)

For second homotopy formula, the Maple commandgem get CL fluxes(multipliers sol,method=“Homotopy2”)

yields divergence expressions in the same form as in (6).

Mathematical Problems in Engineering 3

Table 1: Multipliers and conserved vectors for PDE (14).

Multiplier Fluxes

Λ(1) = 𝑢𝑡

𝜙(1)= 𝜆2+1

2𝑢𝑢𝑥𝑥+1

2𝑢𝑢𝑧𝑧−1

2𝑢2

𝑡− 𝜆2 cosh 𝑢

𝜓(1) = −1

2𝑢𝑢𝑡𝑥+1

2𝑢𝑡𝑢𝑥

𝜋(1) = −1

2𝑢𝑢𝑡𝑧+1

2𝑢𝑡𝑢𝑧

Λ(2) = 𝑢𝑥

𝜙(2)=1

2𝑢𝑢𝑡𝑥−1

2𝑢𝑡𝑢𝑥

𝜓(2)= 𝜆2+1

2𝑢𝑢𝑧𝑧+1

2𝑢2

𝑥− 𝜆2 cosh 𝑢 − 1

2𝑢𝑢𝑡𝑡

𝜋(2) = −1

2𝑢𝑢𝑥𝑧+1

2𝑢𝑥𝑢𝑧

Λ(3) = 𝑢𝑧

𝜙(3)=1

2𝑢𝑢𝑡𝑧−1

2𝑢𝑡𝑢𝑧

𝜓(3) = −1

2𝑢𝑢𝑥𝑧+1

2𝑢𝑥𝑢𝑧

𝜋(3) = 𝜆2 +1

2𝑢𝑢𝑥𝑥+1

2𝑢2𝑧− 𝜆2 cosh 𝑢 − 1

2𝑢𝑢𝑡𝑡

Λ(4) = 𝑥𝑢𝑡+ 𝑡𝑢𝑥

𝜙(4)=1

2𝑢𝑢𝑥−1

2𝑥𝑢2

𝑡+1

2𝑡𝑢𝑢𝑡𝑥−1

2𝑡𝑢𝑡𝑢𝑥+1

2𝑥𝑢𝑢𝑧𝑧+1

2𝑥𝑢𝑢𝑥𝑥− 𝜆2𝑥 cosh 𝑢 + 𝜆2𝑥

𝜓(4)= −1

2𝑥𝑢𝑢𝑡𝑥−1

2𝑡𝑢𝑢𝑡𝑡+1

2𝑥𝑢𝑥𝑢𝑡+1

2𝑡𝑢𝑢𝑧𝑧+ 𝜆2𝑡 − 𝜆2𝑡 cosh 𝑢 − 1

2𝑢𝑢𝑡+1

2𝑡𝑢2

𝑥

𝜋(4)= −1

2𝑥𝑢𝑢𝑡𝑧+1

2𝑥𝑢𝑡𝑢𝑧−1

2𝑡𝑢𝑢𝑥𝑧+1

2𝑡𝑢𝑥𝑢𝑧

Λ(5)= −𝑧𝑢

𝑥+ 𝑥𝑢𝑧

𝜙(5) =1

2𝑧𝑢𝑢𝑡𝑥+1

2𝑥𝑢𝑢𝑡𝑧+1

2𝑧𝑢𝑡𝑢𝑥−1

2𝑥𝑢𝑡𝑢𝑧

𝜓(5)= −1

2𝑢𝑢𝑧−1

2𝑧𝑢𝑢𝑧𝑧−1

2𝑥𝑢𝑢𝑥𝑧+1

2𝑧𝑢𝑢𝑡𝑡−1

2𝑧𝑢2

𝑥+1

2𝑥𝑢𝑥𝑢𝑧− 𝜆2𝑧 + 𝜆2𝑧 cosh 𝑢

𝜋(5)=1

2𝑢𝑢𝑥+1

2𝑥𝑢𝑢𝑥𝑥− 𝜆2𝑥 cosh 𝑢 + 1

2𝑥𝑢2

𝑧+1

2𝑧𝑢𝑢𝑥𝑧−1

2𝑧𝑢𝑧𝑢𝑥−1

2𝑥𝑢𝑢𝑡𝑡+ 𝜆2𝑥

Λ(6)= 𝑡𝑢𝑧+ 𝑧𝑢𝑡

𝜙(6) =1

2𝑢𝑢𝑧−1

2𝑧𝑢2𝑡+1

2𝑡𝑢𝑢𝑡𝑧−1

2𝑡𝑢𝑡𝑢𝑧+1

2𝑧𝑢𝑢𝑧𝑧+1

2𝑧𝑢𝑢𝑥𝑥+ 𝜆2𝑧 − 𝜆2𝑧 cosh 𝑢

𝜓(6) = −1

2𝑧𝑢𝑢𝑡𝑥+1

2𝑧𝑢𝑡𝑢𝑥−1

2𝑡𝑢𝑢𝑥𝑧+1

2𝑡𝑢𝑥𝑢𝑧

𝜋(6) = −1

2𝑡𝑢𝑢𝑡𝑡+ 𝜆2𝑡 − 𝑡𝜆2 cosh 𝑢 − 1

2𝑢𝑢𝑡+1

2𝑡𝑢2𝑧−1

2𝑧𝑢𝑢𝑡𝑧+1

2𝑧𝑢𝑧𝑢𝑡+1

2𝑡𝑢𝑢𝑥𝑥

The PDE (1) has no scaling symmetry; therefore, we can-not apply the scaling symmetry formula here for derivationof fluxes.

2.2. Lubrication Models. Now we will study two lubricationmodels for conservation laws point of view. Gandarias andMedina [23] performed the symmetry analysis of lubricationmodel

𝑢𝑡= 𝑓 (𝑢) 𝑢

𝑥𝑥𝑥𝑥, (7)

where 𝑓 is an arbitrary function. For 𝑓(𝑢) = 𝑐(𝑢 + 𝑏)𝑎 and𝑓(𝑢) = 𝛾𝑒𝛼𝑢, this equation has some extra symmetry [23].Without loss of generality, take 𝑓(𝑢) = 𝑢 + 𝑏 in (7); we have

𝑢𝑡= (𝑢 + 𝑏) 𝑢

𝑥𝑥𝑥𝑥, (8)

where 𝑏 is arbitrary constant. Consider themultipliers of formΛ(𝑡, 𝑥, 𝑢) in GeM Maple routines, and then we obtain thefollowing four multipliers:

Λ(1)(𝑡, 𝑥, 𝑢) =

1

(𝑢 + 𝑏), Λ

(2)(𝑡, 𝑥, 𝑢) =

𝑥

(𝑢 + 𝑏),

Λ(3)(𝑡, 𝑥, 𝑢) =

𝑥2

2 (𝑢 + 𝑏), Λ

(4)(𝑡, 𝑥, 𝑢) =

𝑥3

6 (𝑢 + 𝑏).

(9)

The fluxes associated with the multipliers given in (7) arecomputed by homotopy first method and are given by

𝜙(1)= ln(𝑢 + 𝑏

𝑏) , 𝜓

(1)= 𝑢𝑥𝑥𝑥,

𝜙(2)= 𝑥 ln(𝑢 + 𝑏

𝑏) , 𝜓

(2)= −𝑢𝑥𝑥+ 𝑥𝑢𝑥𝑥𝑥,

𝜙(3)=1

2𝑥2 ln(𝑢 + 𝑏𝑏) , 𝜓

(3)= 𝑢𝑥− 𝑥𝑢𝑥𝑥+𝑥2𝑢𝑥𝑥𝑥

2,

4 Mathematical Problems in Engineering

Table 2: Multipliers and conserved vectors for PDE (15).

Multiplier Fluxes

Λ(1) = 1𝜙(1)= 𝑢

𝜓(1) = −𝑢𝑥𝑥𝑥𝑥−5

3𝑢3 − 5𝑢𝑢

𝑥𝑥−15

4𝑢2𝑥

Λ(2) = 2𝑢2 + 𝑢𝑥𝑥

𝜙(2) =2

3𝑢3 +1

2𝑢𝑢𝑥𝑥

𝜓(2) = 4𝑢𝑢𝑥𝑢𝑥𝑥𝑥+1

2𝑢𝑡𝑢𝑥− 4𝑢2𝑥𝑢𝑥𝑥−9

2𝑢𝑢2𝑥𝑥+1

2𝑢2𝑥𝑥𝑥− 2𝑢2𝑢

𝑥𝑥𝑥𝑥− 𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥−1

2𝑢𝑢𝑡𝑥− 10𝑢3𝑢

𝑥𝑥− 2𝑢5

Λ(3) = 𝑥 + 5𝑡𝑢2 +5

2𝑡𝑢𝑥𝑥

𝜙(3)=5

3𝑡𝑢3+5

4𝑢𝑢𝑥𝑥+ 𝑥𝑢

𝜓(3) = 𝑢𝑥𝑥𝑥− 𝑥𝑢𝑥𝑥𝑥𝑥+5

4𝑡𝑢2𝑥𝑥𝑥−5

4𝑡𝑢𝑢𝑡𝑥− 5𝑡𝑢2𝑢

𝑥𝑥𝑥𝑥−5

2𝑡𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥+ 10𝑡𝑢𝑢

𝑥𝑢𝑥𝑥𝑥+5

4𝑡𝑢𝑡𝑢𝑥−5

3𝑥𝑢3

−5𝑡𝑢5−15

4𝑥𝑢4

𝑥− 10𝑡𝑢

2

𝑥𝑢𝑥𝑥−45

4𝑡𝑢𝑢2

𝑥𝑥− 5𝑥𝑢𝑢

𝑥𝑥− 25𝑡𝑢

3𝑢𝑥𝑥+15

4𝑢𝑢𝑥

Λ(4) = 𝑢𝑢𝑥𝑥+1

2𝑢2𝑥+1

6𝑢𝑥𝑥𝑥𝑥+4

9𝑢3

𝜙(4) =1

9𝑢4 +1

3𝑢2𝑢𝑥𝑥+1

6𝑢𝑢2𝑥+1

12𝑢𝑢𝑥𝑥𝑥𝑥

𝜓(4) = −5

6𝑢3𝑢2𝑥+1

12𝑢𝑢2𝑥𝑥𝑥−10

27𝑢6 −11

16𝑢4𝑥−1

12𝑢2𝑥𝑥𝑥𝑥−1

12𝑢𝑢𝑡𝑥𝑥𝑥+1

12𝑢𝑥𝑢𝑡𝑥𝑥−1

2𝑢2𝑥𝑢𝑥𝑥𝑥𝑥

−1

12𝑢𝑡𝑥𝑢𝑥𝑥+ 𝑢𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥+1

12𝑢𝑡𝑢𝑥𝑥𝑥−1

3𝑢2𝑢𝑡𝑥−1

12𝑢𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥+1

3𝑢𝑢𝑡𝑢𝑥+1

36𝑢3𝑥𝑥

−11

4𝑢2𝑢2

𝑥𝑥−20

9𝑢4𝑢𝑥𝑥−4

9𝑢3𝑢𝑥𝑥𝑥𝑥−7

2𝑢𝑢2

𝑥𝑢𝑥𝑥+1

2𝑢2𝑢𝑥𝑢𝑥𝑥𝑥

𝜙(4)=1

6𝑥3 ln(𝑢 + 𝑏𝑏) ,

𝜓(4)= −𝑢 + 𝑥𝑢

𝑥−𝑥2𝑢𝑥𝑥

2+𝑥3𝑢𝑥𝑥𝑥

6.

(10)We will get the same fluxes for (7) if we define higher ordermultipliers in GeMMaple routines.

Another interesting lubrication model is

𝑢𝑡+𝑢𝑥𝑥𝑥𝑥

𝑒𝑢= 0. (11)

It is obtained by taking 𝑓(𝑢) = 𝑒−𝑢 in (7). The GeM Mapleroutines yield the following fourmultipliers of formΛ(𝑡, 𝑥, 𝑢):

Λ(1)(𝑡, 𝑥, 𝑢) = 𝑒

𝑢, Λ

(2)(𝑡, 𝑥, 𝑢) = 𝑥𝑒

𝑢,

Λ(3)(𝑡, 𝑥, 𝑢) =

1

2𝑥2𝑒𝑢, Λ

(4)(𝑡, 𝑥, 𝑢) =

1

6𝑥3𝑒𝑢.

(12)

The corresponding fluxes obtained by homotopy firstmethod are

𝜙(1)= −1 + 𝑒

𝑢, 𝜓

(1)= 𝑢𝑥𝑥𝑥,

𝜙(2)= 𝑥 (−1 + 𝑒

𝑢) , 𝜓

(2)= −𝑢𝑥𝑥+ 𝑥𝑢𝑥𝑥𝑥,

𝜙(3)=1

2𝑥2(−1 + 𝑒

2) , 𝜓

(3)= 𝑢𝑥− 𝑥𝑢𝑥𝑥+𝑥2𝑢𝑥𝑥𝑥

2,

𝜙(4)=1

6𝑥3(−1 + 𝑒

𝑢) ,

𝜓(4)= −𝑢 + 𝑥𝑢

𝑥−𝑥2𝑢𝑥𝑥

2+𝑥3𝑢𝑥𝑥𝑥

6.

(13)

The conservation laws fluxes derived here can be used to findthe solution of lubrication models and will be considered infuture work.

2.3. sinh-Poisson Equation. The (2 + 1)-dimensional sinh-Poisson equation is [24]

𝑢𝑥𝑥+ 𝑢𝑧𝑧− 𝑢𝑡𝑡= 𝜆2 sinh 𝑢, (14)

where 𝑢(𝑡, 𝑥, 𝑧). The conservation laws for PDE (14) arederived here by usingGeM routines. Consider themultipliersof formΛ(𝑡, 𝑥, 𝑢, 𝑢

𝑡, 𝑢𝑥, 𝑢𝑧) in GeM routines, then it will yield

six multipliers not containing any arbitrary function. Theexpression for fluxes is computed by using first homotopyformula. The multipliers and associated conserved vectorscomputed by first homotopy formula are given in Table 1.

2.4. Kaup-Kupershmidt Equation. Now, we will compute theconservation laws for the fifth order Kaup-Kupershmidt [25]:

𝑢𝑡= 𝑢𝑥𝑥𝑥𝑥𝑥+ 5𝑢𝑢

𝑥𝑥𝑥+25

2𝑢𝑥𝑢𝑥𝑥+ 5𝑢2𝑢𝑥. (15)

The GeM Maple routines yield three multipliers of theform Λ(𝑡, 𝑥, 𝑢, 𝑢

𝑡, 𝑢𝑥, 𝑢𝑥𝑥) for PDE (14). The first homo-

topy formula is applied to derive the expressions for con-servation laws fluxes. One more multiplier can be com-puted if we consider higher order multipliers of the formΛ(𝑡, 𝑥, 𝑢, 𝑢

𝑡, 𝑢𝑥, 𝑢𝑥𝑥, 𝑢𝑥𝑥𝑥, 𝑢𝑥𝑥𝑥𝑥). All themultipliers and asso-

ciated conserved vectors for PDE (14) computed by firsthomotopy formula are presented in Table 2.

Mathematical Problems in Engineering 5

Table 3: Multipliers and conserved vectors for PDE (16).

Multiplier Fluxes

Λ(1) = 1 𝜙(1)= 𝑢, 𝜓(1) = 5𝑢

𝑥𝑢𝑥𝑥− 𝑢𝑥𝑥𝑥𝑥+ 5𝑢2𝑢𝑥𝑥+ 5𝑢𝑢

2

𝑥− 𝑢5

Λ(2) = 𝑢𝜙(2)=1

2𝑢2,

𝜓(2)= 5𝑢𝑢

𝑥𝑢𝑥𝑥−1

2𝑢2

𝑥𝑥+ 𝑢𝑥𝑢𝑥𝑥𝑥−5

3𝑢3

𝑥+5

2𝑢2𝑢2

𝑥+ 5𝑢3𝑢𝑥𝑥−5

6𝑢6− 𝑢𝑢𝑥𝑥𝑥𝑥

Λ(3) = 𝑢5 + 𝑢𝑥𝑥𝑥𝑥− 5𝑢𝑥𝑢𝑥𝑥

−5𝑢2𝑢𝑥𝑥− 5𝑢𝑢2

𝑥

𝜙(3)=1

6𝑢6−5

4𝑢3𝑢𝑥𝑥−5

3𝑢𝑢𝑥𝑢𝑥𝑥+1

2𝑢𝑢𝑥𝑥𝑥𝑥

𝜓(3) = −1

2𝑢2𝑥𝑥𝑥𝑥−1

2𝑢10 − 25𝑢𝑢3

𝑥𝑢𝑥𝑥−25

2𝑢2𝑥𝑢2𝑥𝑥−25

2𝑢2𝑢4𝑥+1

2𝑢𝑡𝑢𝑥𝑥𝑥

−5

3𝑢𝑡𝑢2

𝑥−5

4𝑢2𝑢𝑡𝑢𝑥− 𝑢5𝑢𝑥𝑥𝑥𝑥+5

3𝑢𝑢𝑥𝑢𝑡𝑥− 25𝑢

3𝑢2

𝑥𝑢𝑥𝑥+ 5𝑢5𝑢𝑥𝑢𝑥𝑥

+5𝑢2𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥+ 5𝑢𝑢2

𝑥𝑢𝑥𝑥𝑥𝑥−1

2𝑢𝑢𝑡𝑥𝑥𝑥+5

4𝑢3𝑢𝑡𝑥+ 5𝑢7𝑢

𝑥𝑥+ 5𝑢6𝑢2

𝑥

+5𝑢𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥− 25𝑢2𝑢

𝑥𝑢2𝑥𝑥+1

2𝑢𝑥𝑢𝑡𝑥𝑥−1

2𝑢𝑡𝑥𝑢𝑥𝑥−25

2𝑢4𝑢2𝑥𝑥

Λ(4) = 5𝑡𝑢2𝑢𝑥𝑥− 25𝑡𝑢2𝑢

𝑥𝑥+ 𝑥𝑢

−25𝑡𝑢𝑥𝑢𝑥𝑥+ 5𝑡𝑢

𝑥𝑥𝑥𝑥

𝜙(4) =1

6𝑡𝑢6 −25

4𝑡𝑢2𝑢2𝑥−25

4𝑡𝑢3𝑢𝑥𝑥−25

3𝑡𝑢𝑢𝑥𝑢𝑥𝑥+5

2𝑡𝑢𝑢𝑥𝑥𝑥𝑥+1

2𝑥𝑢2

𝜓(4)= −25

4𝑡𝑢𝑡𝑢𝑥+5

4𝑢3𝑢𝑥− 5𝑡𝑢

5𝑢𝑥𝑥𝑥𝑥+1

2𝑢𝑥𝑢𝑥𝑥+ 25𝑡𝑢𝑢

2

𝑥𝑢𝑥𝑥𝑥𝑥+ 25𝑡𝑢

2𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥+ 5𝑥𝑢𝑢

𝑥𝑢𝑥𝑥

−125𝑡𝑢3𝑢2𝑥𝑢𝑥𝑥− 125𝑡𝑢2𝑢

𝑥𝑢2𝑥𝑥− 125𝑡𝑢𝑢3

𝑥𝑢𝑥𝑥−1

2𝑥𝑢2𝑥𝑥−5

6𝑥𝑢6 −5

2𝑡𝑢2𝑥𝑥𝑥𝑥−5

3𝑥𝑢3𝑥+5

3𝑢𝑢2𝑥

−3

2𝑢𝑢𝑥𝑥𝑥+ 25𝑡𝑢5𝑢

𝑥𝑢𝑥𝑥+ 25𝑡𝑢

𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥−5

2𝑡𝑢10 −125

2𝑡𝑢2𝑥𝑢2𝑥𝑥−25

3𝑡𝑢𝑡𝑢2𝑥+5

2𝑡𝑢𝑡𝑢𝑥𝑥𝑥

+5

2𝑡𝑢𝑥𝑢𝑡𝑥𝑥+ 25𝑡𝑢

6𝑢2

𝑥+ 25𝑡𝑢

7𝑢𝑥𝑥−5

2𝑡𝑢𝑡𝑥𝑢𝑥𝑥−5

2𝑡𝑢𝑢𝑡𝑥𝑥𝑥+25

4𝑡𝑢3𝑢𝑡𝑥

+𝑥𝑢𝑥𝑢𝑥𝑥𝑥+ 5𝑥𝑢3𝑢

𝑥𝑥− 𝑥𝑢𝑢

𝑥𝑥𝑥𝑥−125

2𝑡𝑢4𝑢𝑥𝑥−125

2𝑡𝑢2𝑢4𝑥+5

2𝑥𝑢2𝑢2𝑥+25

3𝑡𝑢𝑢𝑥𝑢𝑡𝑥

Λ(5) = −4

3𝑢7 + 9𝑢4𝑢

𝑥𝑥+ 8𝑢3𝑢2

𝑥

−2𝑢2𝑢𝑥𝑥𝑥𝑥+ 14𝑢

2𝑢𝑥𝑢𝑥𝑥− 𝑢𝑢2

𝑥𝑥

+2𝑢𝑢𝑥𝑢𝑥𝑥+28

3𝑢𝑢3𝑥+ 7𝑢2𝑥𝑢𝑥𝑥

+𝑢𝑥𝑥𝑢𝑥𝑥𝑥− 2𝑢𝑥𝑢𝑥𝑥𝑥𝑥+ 𝑢𝑡𝑥

𝜙(5)= −1

6𝑢8+1

2𝑢4𝑢2

𝑥+3

2𝑢5𝑢𝑥𝑥+14

5𝑢3𝑢𝑥𝑢𝑥𝑥+28

15𝑢2𝑢3

𝑥+5

4𝑢4

𝑥

+27

4𝑢𝑢2𝑥𝑢𝑥𝑥+7

4𝑢2𝑢𝑥𝑢𝑥𝑥𝑥−1

4𝑢2𝑢𝑥𝑥−1

2𝑢3𝑢𝑥𝑥𝑥𝑥+1

3𝑢𝑢𝑥𝑥𝑢𝑥𝑥𝑥

−2

3𝑢𝑢𝑥𝑢𝑥𝑥𝑥𝑥+5

3𝑢𝑥𝑢2𝑥𝑥+5

3𝑢2𝑥𝑢𝑥𝑥𝑥+1

2𝑢𝑢𝑡𝑥+1

2𝑢𝑡𝑢𝑥−1

2𝑢𝑥𝑢𝑥𝑥𝑥𝑥𝑥

𝜓(5) =2

3𝑢𝑢𝑡𝑥𝑢𝑥𝑥𝑥+1

3𝑢3𝑥𝑥𝑥+19

18𝑢6𝑥+1

2𝑢4𝑥𝑥− 8𝑢3𝑢2

𝑥𝑢𝑥𝑥𝑥𝑥− 𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑢𝑥𝑥𝑥𝑥

−9𝑢4𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥− 7𝑢2

𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥+140

3𝑢3𝑢3

𝑥𝑢𝑥𝑥−20

3𝑢7𝑢𝑥𝑢𝑥𝑥−28

3𝑢𝑢3

𝑥𝑢𝑥𝑥𝑥𝑥

+𝑢𝑢2𝑥𝑥𝑢𝑥𝑥𝑥𝑥+1

4𝑢𝑡𝑢3𝑥+ 16𝑢𝑢2

𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥+ 12𝑢3𝑢

𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥+1

2𝑢𝑥𝑢𝑡𝑥𝑥𝑥𝑥− 2𝑢𝑢

𝑥𝑢𝑥𝑥𝑥𝑢𝑥𝑥𝑥𝑥

−14𝑢2𝑢𝑥𝑢𝑥𝑥𝑢𝑥𝑥𝑥𝑥−13

4𝑢2𝑢𝑥𝑢𝑡𝑥𝑥−1

2𝑢2𝑢𝑡𝑢𝑥𝑥𝑥+133

6𝑢6𝑢2𝑥𝑥+ 10𝑢4𝑢4

𝑥−7

3𝑢2𝑥𝑢𝑡𝑥𝑥

+𝑢𝑡𝑢2

𝑥𝑥−1

2𝑢𝑥𝑥𝑢𝑡𝑥𝑥𝑥+1

2𝑢𝑡𝑥𝑥𝑢𝑥𝑥𝑥−1

2𝑢𝑡𝑥𝑢𝑥𝑥𝑥𝑥− 𝑢𝑢𝑥𝑥𝑢𝑡𝑥𝑥+86

3𝑢𝑢𝑥𝑥𝑢4

𝑥+ 14𝑢

2𝑢3

𝑥𝑢𝑥𝑥𝑥

+3𝑢𝑥𝑢2

𝑥𝑥𝑢𝑥𝑥𝑥− 5𝑢𝑢

𝑥𝑢3

𝑥𝑥+ 4𝑢2𝑢2

𝑥𝑥𝑢𝑥𝑥𝑥+ 45𝑢

4𝑢𝑥𝑢2

𝑥𝑥− 3𝑢2𝑢𝑥𝑢𝑥𝑥𝑥+ 10𝑢

4𝑢2

𝑥𝑢𝑥𝑥𝑥

+67

2𝑢2𝑢2𝑥𝑢2𝑥𝑥−3

2𝑢5𝑢𝑡𝑥+3

2𝑢4𝑢𝑡𝑢𝑥−1

4𝑢𝑢2𝑥𝑢𝑡𝑥+4

3𝑢7𝑢𝑥𝑥𝑥𝑥−20

3𝑢9𝑢𝑥𝑥

+15

4𝑢2𝑢𝑡𝑥𝑢𝑥𝑥−35

6𝑢8𝑢2

𝑥−1

2𝑢𝑢𝑡𝑡+1

2𝑢3𝑢𝑡𝑥𝑥𝑥+5

9𝑢12 −1

2𝑢𝑢𝑡𝑢𝑥𝑢𝑥𝑥+ 26𝑢

5𝑢2

𝑥𝑢𝑥𝑥

+2

3𝑢6𝑢𝑥𝑢𝑥𝑥𝑥+ 𝑢𝑥𝑢2𝑥𝑥𝑥𝑥+ 𝑢2𝑢2

𝑥𝑥𝑥𝑥−17

3𝑢3𝑢3𝑥𝑥+70

3𝑢2𝑢5𝑥−70

9𝑢6𝑢3𝑥−1

2𝑢2𝑥𝑢2𝑥𝑥𝑥

−1

2𝑢4𝑢2𝑥𝑥𝑥+40

3𝑢3𝑥𝑢2𝑥𝑥−2

3𝑢4𝑥𝑢𝑥𝑥𝑥+7

3𝑢𝑥𝑢𝑡𝑥𝑢𝑥𝑥

+14

5𝑢𝑡𝑢2𝑢2

𝑥−4

3𝑢𝑡𝑢𝑥𝑢𝑥𝑥𝑥+2

3𝑢𝑢𝑥𝑢𝑡𝑥𝑥𝑥−14

5𝑢3𝑢𝑥𝑢𝑡𝑥

6 Mathematical Problems in Engineering

2.5. Modified Sawada-Kotera Equation. Consider the fifthorder modified SK equation:

𝑢𝑡= 𝑢𝑥𝑥𝑥𝑥𝑥− (5𝑢

𝑥𝑢𝑥𝑥+ 5𝑢𝑢

2

𝑥+ 5𝑢2𝑢𝑥𝑥− 𝑢5)𝑥. (16)

For PDE (16), two conserved densities were derived byfirst computing Lax pair (see [26]). The higher orderconservation laws fluxes exist for higher order multipliersand are not reported in [26]. Consider the multipliers ofform Λ(𝑡, 𝑥, 𝑢, 𝑢

𝑡, 𝑢𝑥, 𝑢𝑡𝑡, 𝑢𝑡𝑥, 𝑢𝑥𝑥, 𝑢𝑥𝑥𝑥, 𝑢𝑥𝑥𝑥𝑥) in GeM rou-

tines, then it will yield two simple and three higher ordermultipliers not containing any arbitrary function.The simplemultipliers yield same fluxes as derived in [26], and threenew fluxes corresponding to higher order multipliers arecomputed. The multipliers and associated conserved vectorscomputed by first homotopy formula are listed in Table 3.

3. Conclusions

The conservation laws for the evolution equation, Benjaminequation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equa-tion were derived by using the symbolic software GeM.First of all, we considered the evolution equation, and thecommands for all GeMMaple routines, were explicitly given.The first order multipliers were defined in GeM Mapleroutines and threemultipliers were obtained.The expressionsfor fluxes were computed by direct method and first andsecond homotopy formulas and equivalent expressions forfluxes were obtained. The scaling symmetry method wasnot applicable here as no scaling symmetry exists for thenonlinear evolution equation. The conservation laws fluxesfor the lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equa-tion were derived by the first homotopy formula. For themodified Sawada-Kotera equation, three new fluxes werederived.

The fluxes derived here can be used in constructing thesolutions of underlying PDEs and will be considered in thefuture work.

Conflict of Interests

The authors declare that there is no conflict of interests.

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