# Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

Authors: Tahir Ayaz; Farhad Ali; Wali Khan Mashwani; Israr Ali Khan; Zabidin Salleh; Ikramullah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.


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## Body

## 1. Introduction

Differential equations (DEs) are ubiquitous in modeling an extensive class of physical phenomena involving variation with respect to one or more independent variables. Therefore, DEs are broadly divided into ordinary DEs (ODEs) and partial DEs (PDEs). In different sectors of science and technology, PDEs have played a significant role. PDEs have numerous applications in mathematics, physics, fluid dynamics, mechanics, and physical chemistry. Modeling of PDEs under special conditions and constraints is advantageous in different situations for an effective manipulation of the varying phenomenon. The majority of real-world problems are almost nonlinear in nature, having no analytical solutions. In order to solve nonlinear problems, various approximations and techniques are used to gain high accuracy. In this regard, the approximate symmetry methods play a significant role. We have used the method of approximate Lie symmetry [1, 2], for PDEs to deal with the dynamical system more accurately. In the 1980s, the method of approximate Lie symmetry was developed by Baikov et al. [3, 4]. In obtaining the approximate solutions to such perturbed PDEs, the approximate symmetry method is an effective one. The extension of Lie’s theory was mainly the basic reason behind the development of approximate symmetry, which deals with the systems by introducing small perturbation [5]. Symmetry applications to physical problems play a pivotal role in the development of conservation laws [6, 7]. The widely recognized KdV equation is a mathematical model for the depiction of weak nonlinear long wavelength waves in various branches of engineering and physics. It explains how waves evolve due to comparable effects of weak nonlinearity and dispersion. A perturbed nonlinear wave equation is a class of approximate symmetries which is computed using two newly developed methods. For both methods, the associated invariant solution with the approximate symmetries is constructed. By discussing the advantages and disadvantages of each method, the symmetries and solutions are compared. So, the Lie group technique in finding the exact solution of a differential equation has lost its importance. But an approximate Lie group technique has been implemented and used in various methods for obtaining additional related information of differential equation. Perturbation analysis is one of the techniques which is used particularly for nonlinear systems.This study is framed in the following manner: Section2 is devoted to the development of exact symmetries and exact conservation laws of the KdV equation. The method to handle the approximate part of the KdV equation is developed in Section 3. The method so developed is applied to tackle the approximate part of the KdV equation for different cases and their corresponding conservation laws in Section 4. The work is concluded by describing the highlights in Section 5.


## 2. Exact Symmetries and Conservation Laws of the Korteweg–de Vries (KdV) Equation

The exact symmetries and conservation laws in the current study for the work considered in [8, 9] are worked out as follows:The Korteweg–de Vries (KdV) equation which is a third-order nonlinear partial differential equation is(1)μt−6μμx+μxxx=0.The infinitesimal symmetry operator is(2)X3=ϕ∂∂x+ϱ∂∂t+φ∂∂μ+φt∂∂μt+φx∂∂μx+φxx∂∂μxx+φxt∂∂μxt+φtt∂∂μtt+φxxx∂∂μxxx.Applying this symmetry operator on (1),(3)X3μt−6μμx+μxxx=0,we get(4)φt−6μφx−6φμx+φxxx=0.The expanded form of equation (4) is(5)φt−ϕtμx+φμ−ϱtμt−ϕμμxμt−ϱμμt2−6μφx+φμ−ϕxμx−ϱxμt−ϕμμx2−ϱxμxμt−6φμx+φxxx+3φμxx−ϕxxxμx−ϱxxxμt+3φμμx−3ϕxxμμx2−3ϱμxxμxμt+3φμx−3ϕxxμxx−3ϱxxμxt+φμμμ−ϕμμxμx3−3ϱμμxμx2μt+3φμμ−9ϕμxμxμxx−6ϱxμμxμxt−3ϱμxμtμxx+φμ−3ϕxμxxx−3ϱxμxxt−ϕμμμμx4−ϱμμμμx3μt−6ϕμμμx2μxx−3ϱμμμx2μxt−3ϱμμμxμtμxx−4φμμxμxxx−3ϱμμxμxxt−3ϕμμxx2−3ϱμμxxμxt−ϱμμtμxxx=0.Substituting equation (1) in equation (5), we get(6)φt−ϕtμx+φμ−ϱtμt−ϕμμxμt−ϱμμt2−6φμx−6μφx+φμ−ϕxμx−ϱxμt−ϕμμx2−ϱμμxμt+φxxx+3φxxμ−ϕxxxμx−ϱxxxμt+3φxμμ−3ϕxxμμx2−3ϱxxμμxμt+3φxμ−3ϕxxμxx−3ϱxxμxt+φμμμ−ϕxμμμx3−3ϱxμμμx2μt+3φμμ−9ϕxμμxμxx−6ϱxμμxμxt−3ϱxμμtμxx+6μφμ−3ϕxμx−φμ−3ϕxμt−3ϱxμxxt−ϕμμμμx4−ϱμμμμx3μt−6ϕμμμx2μxx−3ϱμμμx2μxt−3ϱμμμxμtμxx−24μϕμμx2+4ϕμμxμt−3ϱμμxμxxt−3ϕμμxx2−3ϱμμxxμxt−6μϱμμxμxt+ϱμμt2=0.Comparing the coefficients of various terms, we get the coefficients and monomials, as shown in Table1.Table 1
The exact symmetries of the given partial differential equation (PDE).
CoefficientsMonomialsφt−6μφx+φxxx=01−ϕt−6φ−6μφμ−ϕx+3φxxμ−ϕxxx+6μφμ−3ϕx=0μx−ϕμ+6μϱμ−3ϱxxμ+4ϕμ=0μxμtφμ−ϱt+6μϱx−ϱxxx−φμ−3ϕx=0μt−ϱμ+ϱμ=0μt26μϕμ+3φxμμ−3ϕxxμ−24μϕμ=0μx23φxμ−3ϕxx=0μxx−3ϱxx=0μxtφμμμ−ϕxμμ=0μx3−3ϱxμμ=0μx2μt3φμμ−9ϕxμ=0μxμxx−6ϱxμ−6μϱμ=0μxμxt−3ϱxμ=0μtμxx−3ϱx=0μxxtϕμμμ=0μx4ϱμμμ=0μx3μtϕμμ=0μx2μxxϱμμ=0μx2μxtϱμμ=0μxμtμxxϱμ=0μxμxxtϕμ=0μxx2ϱμ=0μxxμxtTable1 yields the required set of PDEs as follows:(7)ϱμ=0,(8)ϕμ=0,(9)ϱx=0,(10)3ϕx−ϱt=0.Form (10),(11)ϕxx=0⇒φxxμ=ϕxxx⇒φxxμ=0,φ=−16ϕt−2μϕx.As(12)ϕx=13ϱt,therefore,(13)φ=−16ϕt−23μϱt,φt=−16ϕtt−23μϱtt,φx=−16ϕxt−23μϱtx,φxxx=0,(14)ϕtt=0,(15)ϱtt=0.Let(16)ϱ=At⇒Attt=0.Integrating twice with respect tot yields(17)⇒Att=k1⇒At=k1t+k2⇒ϱ=k1t+k2.From (10),(18)3ϕx−ϱt=0,ϕμ=0⇒ϕ=Bxt,ϕx=13ϱt⇒ϕx=13k1.Integrating with respect to “x,”(19)⇒ϕ=13k1x+Dt⇒ϕtt=Dttt=0⇒Dt=k3t+k4⇒ϕ=13k1x+k3t+k4.From (13),(20)φ=−16ϕt−23μϱt=−16k3−23k1μ.The general solution is(21)φ=−23k1μ−16k3,ϱ=k1t+k2,ϕ=13k1x+k3t+k4.Hence, the Lie symmetry generators for the KdV equation are given, as shown in Table2.Table 2
Lie symmetry generator of KdV equation.
Lie symmetry generatorsX1=1/3x∂/∂x+t∂/∂t−2/3μ∂/∂μX2=∂/∂tX3=∂/∂X4=∂/∂x


## 3. A New Procedure to Find the Approximate Symmetries

This section explains the development of the method for the approximate symmetries of the KdV equation. The KdV (1) is perturbed with the function fx,t,μx,t,μt,x as(22)μt−6μμx+μxxx+ɛfx,t,μx,t,μt,x=0,where ɛ is a small parameter, causing the required perturbation in the KdV equation. The exact and approximate parts of (22) are(23)Ee=μt−6μμx+μxxx,Ea=fx,t,μx,t.Equation (22) can now be written in a more compact form as(24)Ee+ɛEa=0.On similar footing, we can combine the exact and approximate Lie symmetries as(25)X=Xe+ɛXa.Here,(26)Xe=ϕe∂∂x+ϱe∂∂t+φe∂∂μ,is the exact Lie symmetry generator, and(27)Xa=ϕa∂∂x+ϱa∂∂t+φa∂∂μis the approximate Lie symmetry generator. Furthermore, ϕ, ϱ, and φ are the unknown functions of x,t, and μ, respectively.Now, applying the generatorX on (24), we have(28)Xe+ɛXaEe+ɛEa=0,which yields(29)XeEe+ɛXaEe+XeEa+Oɛ2=0.The comparison of coefficients ofɛ0 and ɛ1, respectively, yields the exact and approximate symmetries of the corresponding PDEs as in the following:(30)XeEe=0,XaEe+XeEa=0.The latter equation additionally gives the approximate Lie symmetries, which will not only provide the approximate conservation laws involved in the dynamics of the KdV equation but will also give the unknown functionfx,t,μx,t,μit,x [8, 10].


## 4. Approximate Symmetries and Corresponding Conservation Laws of the KdV Equation

In this section, we apply the developed method to find out the approximate symmetries. This method is applied and discussed for different cases. Considering the perturbed KdV equation [6, 11, 12],(31)μt−6μμx+μxxx+ɛfx,y,n,t,nt,nx,mt,mx=0.By employing the method developed in [13–15] for the expansion of μ,(32)μ=m+ɛn.Using this expansion in (31),(33)mt+ɛnt−6m+ɛnmx+ɛnx+mxxx+ɛnxxx=ɛfx,y,n,t,nt,nx,mt,mx,mt+ɛnt−6mmx−6ɛmnx−6ɛnmx−6ɛ2nnx+mxxx+ɛnxxx=ɛfx,y,n,t,nt,nx,mt,mx,mt−6mmx+mxxx+ɛnt−6mnx−6nmx+nxxx+ɛ2−6nnx=ɛfx,y,n,t,nt,nx,mt,mx.Equation (33) in more compact form is (neglecting higher power of ϵ)(34)Δe+ɛΔa=0.The comparison of the coefficients ofϵ0 and ϵ1 in (33) gives(35)Δe≔mt−6mmx+mxxx=0,Δa≔nt−6mnx−6nmx+nxxx−fx,y,n,t,nt,nx,mt,mx=0.The Lie symmetry generator is(36)X=Xe+ɛXa=0.Here,(37)Xe=ϕe∂∂x+ϱe∂∂t+φe∂∂m+ϕe∂∂n,Xa=ϕa∂∂x+ϱa∂∂t+φa∂∂m+ϕa∂∂n.Applying the Lie generator,(38)XΔe+ɛΔa=0,Xe+ɛXaΔe+ɛΔa=0,which gives us(39)XeΔe+ɛXaΔe+XeΔa+oɛ2=0,XeΔe=0,XaΔe+XeΔa=0.We now discuss the following cases in a bit detail.Case I.
Let(40)fx,y,n,t,nt,nx,mt,mx=−mt−nt.
Then, determining the system of PDEs from (35),(41)ϱtt=0,ϱm=0,ϕt=0,ϱn=0,ϕn=0,ϕm=0,ϱx=0,ϕx=3ϱt,φ=−23mϱt,ϕ=−23ϱtn.
As(42)∂ϱ∂m=0,∂ϱ∂n=0,∂ϱ∂x=0,which implies that “ϱ” is the function of “t” alone. Therefore,(43)ϱtt=0.
Integrating the above equation twice with respect to “t” yields(44)ϱ=c1t+c2.
Also,(45)∂ϕ∂t=0,∂ϕ∂m=0,∂ϕ∂n=0,which shows that “ϕ” is the function of “x” alone. Therefore,(46)∂ϕ∂x=13ϱt.
Putting the value of “ϱt” in (46), we get(47)∂ϕ∂x=13c1.
Integrating (47), we get(48)ϕ=13c1x+c3.
Now,(49)φ=−23mϱt.
Putting the value of “ϱt” in (49),(50)φ=−23mc1.
By taking(51)ϕ=−23ϱtn,and putting the value of “ϱt” in (51),(52)ϕ=−23c1n.
Therefore,(53)ϕ=13c1x+c3,ϱ=c1t+c2,φ=−23mc1,ϕ=−23c1n.
The corresponding symmetry generators are tabulated in Table3.Table 3
Lie symmetry generators.
Lie symmetry generatorsX1=1/3x∂/∂x+t∂/∂t−2/3∂/∂n−2/3m∂/∂mX2=∂/∂tX3=∂/∂x


### 4.1. Conservation Laws

The conservation laws are developed as in the following:(54)X1ψx,y,n,t,nt,nx,mt,mx=0,13x∂∂x+t∂∂t−23∂∂n−23m∂∂mψ=0,13xψx+tψt−23ψn−23mψm=0,3dxx=dtt=dn−2/3=−32dmm=dψ0.Now, by taking(55)3dxx=dtt⇒x3=c1t⇒c1=x3t,3dxx=dn−2/3⇒lnx3=−23n+c2⇒c2=x3e3/2n,3dxx=−32dmm⇒x3=c3m−32⇒c3=x3m3/2,dtt=dn−2/3⇒lnt=−23n+c4⇒c4=te3/2n,dtt=−32dmm⇒t=c5m−32⇒c5=tm32,dn−2/3=−32dmm⇒n+c6=lnm⇒c6=me−n,so(56)ψ=c1+c2+c3=x3t+x3e3/2n+x3m3/2+te3/2n+tm32+me−n.Furthermore,(57)X2ψx,y,n,t,nt,nx,mt,mx=0,ψt=0⇒ψ=c,X3ψx,y,n,t,nt,nx,mt,mx=0,ψx=0⇒ψ=c.Following are the symmetries and their corresponding conservation laws of Case 1.Case 2.
Let(58)fx,y,n,t,nt,nx,mt,mx=−mx.
From (35), we get after comparing the coefficients of ϵ0 and ϵ1,(59)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−mx=0.
Applying (36) to (59) yields the following system of PDEs:(60)ϕt=0,ϕn=0,ϕm=0,ϕm=0,ϕn=6ϕn+1,ϕx=13ϱt,ϕx=0,ϱm=0,ϕtt=0,ϱn=0,φ=−23ϱtm−16ϕtϱx=0,ϱtt=0.
Solving the above system of PDEs, we get the following results:(61)φ=−23c1m−16c4,ϕ=6c3n+c3,ϱ=c1t+c2,ϕ=13c1x+c4t+c5.
The approximate symmetries and their corresponding conservation laws in this case are given in Table4.Table 4
Lie symmetry generators and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=1/3x∂/∂x+t∂/∂t−2/3m∂/∂mψ1=x3/t+xm3+tm3/2X2=∂/∂tψ2=fx,y,n,nt,nx,mt,mxX3=6n+1∂/∂nψ3=gx,y,t,nt,nx,mt,mxX4=t∂/∂x−1/6∂/∂mψ4=x/t+6mX5=∂/∂xψ5=hy,n,t,nt,nx,mt,mxCase 3.
For this case, take(62)fx,y,n,t,nt,nx,mt,mx=−nx.
From (35), we get after comparing the coefficients of ϵ0 and ϵ1,(63)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−nx=0.
This results in the following equations:(64)ϕt=0,ϕtt=0,ϕm=0,ϕx=0,ϕn=ϕn,ϕn=0,ϕx=0,ϱt=0,ϕm=0,ϱm=0,φ=−16ϕtϱn=0,ϱx=0,φ=−16c1,ϕ=c3n,ϱ=c4,ϕ=c1t+c2.
Following are the symmetries and corresponding conservation laws of this Case 3.Case 4.
For this case, take(65)fx,y,n,t,nt,nx,mt,mx=mn,then the system defined in (35) gives(66)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx+mn=0.
Applying (36) to (66), we get the following set of PDEs:(67)ϱt=0,ϕt=16nϕt,ϕx=0,ϕm=0,ϱx=0,ϕn=ϕn,ϕtt=0,ϕx=0,ϱm=0,ϕm=0,ϱn=0,ϕn=0,φ=−16ϕt.
The above equations yield(68)ϕ=c1t+c2,ϱ=c3,φ=−16c1,ϕ=16nc1t+6c4.
Following are the symmetries and corresponding conservation laws of Case 4.Case 5.
Let(69)fx,y,n,t,nt,nx,mt,mx=−n,then the system defined in (35) gives(70)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−n=0.
Applying (36) to (70) produces the following set of PDEs:(71)ϕt=ϱtn,ϕn=0,ϕm=0,ϕm=0,ϕn=ϕn,ϱtt=0,ϕx=0,ϱm=0,ϕx=13ϱtϱn=0,ϱx=0.
Solving the above equations, we get(72)ϕ=c1x3+tc3+c4,ϱ=c1t+c2,φ=−23mϱtc1−16ϕtc3,ϕ=nc1t+c5.
Following are the symmetries and corresponding conservation laws of Case 5.Case 6.
Assume(73)fx,y,n,t,nt,nx,mt,mx=−nmt,then the system defined in (35) gives(74)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−nmt=0.
Applying (36) to (74) results in the following set of PDEs:(75)ϱtx=0,ϕt=0,ϕx=0,ϕm=0,ϕn=0,ϕn=ϕn,ϕm=0,ϱn=0,ϕx=0,ϱt=0,ϕt=0,ϱm=0.
Solving the above set of equations, we get(76)ϕ=c3,ϱ=c2,φ=0,ϕ=c1n.
Following are the symmetries and corresponding conservation laws of Case 6.


## 4.1. Conservation Laws

The conservation laws are developed as in the following:(54)X1ψx,y,n,t,nt,nx,mt,mx=0,13x∂∂x+t∂∂t−23∂∂n−23m∂∂mψ=0,13xψx+tψt−23ψn−23mψm=0,3dxx=dtt=dn−2/3=−32dmm=dψ0.Now, by taking(55)3dxx=dtt⇒x3=c1t⇒c1=x3t,3dxx=dn−2/3⇒lnx3=−23n+c2⇒c2=x3e3/2n,3dxx=−32dmm⇒x3=c3m−32⇒c3=x3m3/2,dtt=dn−2/3⇒lnt=−23n+c4⇒c4=te3/2n,dtt=−32dmm⇒t=c5m−32⇒c5=tm32,dn−2/3=−32dmm⇒n+c6=lnm⇒c6=me−n,so(56)ψ=c1+c2+c3=x3t+x3e3/2n+x3m3/2+te3/2n+tm32+me−n.Furthermore,(57)X2ψx,y,n,t,nt,nx,mt,mx=0,ψt=0⇒ψ=c,X3ψx,y,n,t,nt,nx,mt,mx=0,ψx=0⇒ψ=c.Following are the symmetries and their corresponding conservation laws of Case 1.Case 2.
Let(58)fx,y,n,t,nt,nx,mt,mx=−mx.
From (35), we get after comparing the coefficients of ϵ0 and ϵ1,(59)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−mx=0.
Applying (36) to (59) yields the following system of PDEs:(60)ϕt=0,ϕn=0,ϕm=0,ϕm=0,ϕn=6ϕn+1,ϕx=13ϱt,ϕx=0,ϱm=0,ϕtt=0,ϱn=0,φ=−23ϱtm−16ϕtϱx=0,ϱtt=0.
Solving the above system of PDEs, we get the following results:(61)φ=−23c1m−16c4,ϕ=6c3n+c3,ϱ=c1t+c2,ϕ=13c1x+c4t+c5.
The approximate symmetries and their corresponding conservation laws in this case are given in Table4.Table 4
Lie symmetry generators and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=1/3x∂/∂x+t∂/∂t−2/3m∂/∂mψ1=x3/t+xm3+tm3/2X2=∂/∂tψ2=fx,y,n,nt,nx,mt,mxX3=6n+1∂/∂nψ3=gx,y,t,nt,nx,mt,mxX4=t∂/∂x−1/6∂/∂mψ4=x/t+6mX5=∂/∂xψ5=hy,n,t,nt,nx,mt,mxCase 3.
For this case, take(62)fx,y,n,t,nt,nx,mt,mx=−nx.
From (35), we get after comparing the coefficients of ϵ0 and ϵ1,(63)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−nx=0.
This results in the following equations:(64)ϕt=0,ϕtt=0,ϕm=0,ϕx=0,ϕn=ϕn,ϕn=0,ϕx=0,ϱt=0,ϕm=0,ϱm=0,φ=−16ϕtϱn=0,ϱx=0,φ=−16c1,ϕ=c3n,ϱ=c4,ϕ=c1t+c2.
Following are the symmetries and corresponding conservation laws of this Case 3.Case 4.
For this case, take(65)fx,y,n,t,nt,nx,mt,mx=mn,then the system defined in (35) gives(66)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx+mn=0.
Applying (36) to (66), we get the following set of PDEs:(67)ϱt=0,ϕt=16nϕt,ϕx=0,ϕm=0,ϱx=0,ϕn=ϕn,ϕtt=0,ϕx=0,ϱm=0,ϕm=0,ϱn=0,ϕn=0,φ=−16ϕt.
The above equations yield(68)ϕ=c1t+c2,ϱ=c3,φ=−16c1,ϕ=16nc1t+6c4.
Following are the symmetries and corresponding conservation laws of Case 4.Case 5.
Let(69)fx,y,n,t,nt,nx,mt,mx=−n,then the system defined in (35) gives(70)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−n=0.
Applying (36) to (70) produces the following set of PDEs:(71)ϕt=ϱtn,ϕn=0,ϕm=0,ϕm=0,ϕn=ϕn,ϱtt=0,ϕx=0,ϱm=0,ϕx=13ϱtϱn=0,ϱx=0.
Solving the above equations, we get(72)ϕ=c1x3+tc3+c4,ϱ=c1t+c2,φ=−23mϱtc1−16ϕtc3,ϕ=nc1t+c5.
Following are the symmetries and corresponding conservation laws of Case 5.Case 6.
Assume(73)fx,y,n,t,nt,nx,mt,mx=−nmt,then the system defined in (35) gives(74)mt−6mmx+mxxx=0,nt−6mnx−6nmx+nxxx−nmt=0.
Applying (36) to (74) results in the following set of PDEs:(75)ϱtx=0,ϕt=0,ϕx=0,ϕm=0,ϕn=0,ϕn=ϕn,ϕm=0,ϱn=0,ϕx=0,ϱt=0,ϕt=0,ϱm=0.
Solving the above set of equations, we get(76)ϕ=c3,ϱ=c2,φ=0,ϕ=c1n.
Following are the symmetries and corresponding conservation laws of Case 6.


## 5. Conclusion

The KdV equation is a 3rd order nonlinear partial differential equation which is modeled for waves on the surface of shallow water. It admits four Lie symmetries given in Table2. In this paper, approximate symmetry techniques are used for finding some classes of the KdV equations that admit more symmetries as compared to the exact KdV equations. We perturbed the KdV equation by different particular functions and found the corresponding Lie symmetries. We found two important classes for the perturbed KdV equation that admits five Lie symmetries. The Lie symmetries along with their conservation laws are given in Tables 2, 4 and 5. In both the tables, we have an extra symmetry which corresponds to an extra conservation law. This extra conservation law is an extra information hidden in the system, the perturbation procedure explored it. Sometimes, the symmetry does not exist for the exact equation, but perturbation enables the equation to admit a symmetry. We saw this phenomenon in this research work by comparing Tables 2–6. Table 1 contains the determining PDEs which provide the set of Lie symmetries admitted by the given PDE. We have 4 Lie symmetries given in Table 2 for exact PDE, while Tables 3 and 6 contain only three Lie symmetries; in these cases, we lose one symmetry (one conservation law). Tables 7 and 8 consist of four Lie symmetries which means that all the conservation laws are recovered in these cases. Table 9 includes the lie symmetry generators and corresponding conservation laws.Table 5
Lie symmetry generators and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=1/3x∂/∂x+t∂/∂t+nt∂/∂n−2/3m∂/∂mψ1=x3/t+x3+x3m3/2+etn+tm3/2X2=∂/∂tψ2=fx,y,n,t,nt,nx,mt,mxX3=∂/∂x−1/6∂/∂mψ3=x+6mX4=∂/∂xψ4=gy,n,t,nt,nx,mt,mxX5=n∂/∂nψ5=hx,y,t,nt,nx,mt,mxTable 6
Lie symmetry generator and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=1/3x∂/∂x+t∂/∂t−2/3∂/∂n−2/3m∂/∂mψ1=x3/t+x3e3/2n+x3m3/2+te3/2n+tm3/2+me−nX2=∂/∂tψ2=fx,y,n,nt,nx,mt,mxX3=∂/∂xψ3=gy,n,t,nt,nx,mt,mxTable 7
Lie symmetry generators and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=−1/6∂/∂x+t∂/∂tψ1=e−6x/tX2=x∂/∂xψ2=fy,n,t,nt,nx,mt,mxX3=n∂/∂nψ3=gx,y,t,nt,nx,mt,mxX4=∂/∂tψ4=hx,y,n,nt,nx,mt,mxTable 8
Lie symmetry generator and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=t∂/∂x+1/6nt∂/∂n−1/6∂/∂mψ1=ex/t/n6+nem+x/t+6mX2=∂/∂xψ2=fy,n,t,nt,nx,mt,mxX3=∂/∂tψ3=gx,y,n,nt,nx,mt,mxX4=n∂/∂nψ4=hx,y,t,nt,nx,mt,mxTable 9
Lie symmetry generators and corresponding conservation laws.
Lie symmetry generatorsCorresponding conservation lawsX1=n∂/∂nψ1=fx,y,t,nt,nx,mt,mxX2=∂/∂tψ2=gx,y,n,nt,nx,mt,mxX3=∂/∂xψ3=hy,n,t,nt,nx,mt,mx



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*Source: 7710333-2021-06-22.xml*