# Lie Symmetry Analysis for the General Classes of Generalized Modified Kuramoto-Sivashinsky Equation

Authors: Rong Qi; Muhammad Mobeen Munir; Nazish Younas; Muhammad Idrees; Jia-Bao Liu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Lie symmetry analysis of differential equations proves to be a powerful tool to solve or at least reduce the order and nonlinearity of the equation. Symmetries of differential equations is the most significant concept in the study of DE’s and other branches of science like physics and chemistry. In this present work, we focus on Lie symmetry analysis to find symmetries of some general classes of KS-type equation. We also compute transformed equivalent equations and some invariant solutions of this equation.


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## Body

## 1. Introduction

Symmetry has been a source of inspiration as a powerful tool in the formulation of the laws of the universe. A great number of physical phenomena is transformed into differential equations. Lie symmetry analysis can change the given differential equation into an equivalent form which is easier to solve. In the analysis of differential equations, the symmetry group approach is quite useful. Galois’s use of finite groups to solve algebraic equations of degrees two, three, and four, as well as to prove that the general polynomial equation of degrees larger than four could not be solved by radicals, served as the paradigm for this application [1–7]. The symmetry group approach is well-known for its importance in the field of differential equations analysis. Sophus Lie is credited with the invention of group categorization methods and the theoretical basis for the Lie groups.There are many different methods for computing the symmetries of differential equations. But Lie symmetry analysis is the best because it is a systematic and algorithmic procedure that does not take into account any guesses or approximations. The principal paper on Lie symmetry is [1], in which Lie demonstrated that a linear 2D, 2nd-order PDE admits at most three boundary invariance group. He processed the maximal invariance group of the one-dimensional heat conductivity and used this analysis to compute its explicit solutions. Symmetry reduction is a leading strategy for resolving nonlinear PDEs. Ovisiannikov made a substantial contribution in persisting with these techniques. He presented the strategy of partially invariant solutions [2, 3]. In this work, he gave a methodology that is based on the idea of group called the equivalence group. Gazizov and Ibragimov [8] tracked down the total symmetry analysis of the one-dimensional Black-Scholes model. Shu-Yong and Feng-Xiang, [9] discussed about the connection between the form invariance and Lie symmetry of nonholonomic framework. Buckwar and Luchko [4] initiated the study of symmetry group of scaling transformation for PDEs of fractional order. Yan et al. [6] performed Lie symmetry analysis and fundamental similarity reductions for the coupled Kuramoto-Sivashinsky(KS) equations. Bozhkov and Dimas [10] computed the conversation laws and group classification for generalized 2D KS equation. Nadjafikhah and Ahangari [7] determined the Lie symmetries and reduction for the two-dimensional damped Kuramoto-Sivashinsky ((2D) DKS) equation. Najafikhah and Ahangari also computed Lie symmetry of 2D generalized Kuramoto-Sivashinsky (KS) equation in [11]. The one-dimensional modified KS-type equation is
(1)ut+uxx+uxxxx+λ−1ux2−σuxx2=0,Chou [12] determined the solution of the cauchy problem for the MKS equation and also computed the solvability with the help of the blow up theorem.In the present paper, we deal with the generalized modified one-dimensional Kuramoto-Sivashinsky (GMKS) type equation and determine the symmetry algebra by using Lie symmetry analysis. In particular, we want to find the optimal system and similarity solutions corresponding to some special cases of GMKS equation. The GMKS type equation is given as(2)fuut+uxx+uxxxx+λ−1ux2−σuxx2=0.We seek the Lie symmetry algebras for this GMKS equations forfu=un, fu=enu, and fu=eun where λ and σ are arbitrary constant and λ≠1. For λ=1, equation (2) is proposed in [13]. Its second derivative satisfies an equation of Cahn-Hilliard type in [14]. This equation has various applications as physical models in biofluids, mechanics, and liquids. In equation (2), u is the velocity function, x is space parameter, and t is time variable. This equation can also be derived from a model in the continuity equation by fitting a suitable function [15]. Actually, the Kuramoto-Sivashinsky equation gives the change of the position of a flame front (Figure 1). It shows the flame front position against time for horizontally propagating methane flame, the movement of a fluid going down a vertical wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium [16]. This equation is also helpful to display solitary pulses in a falling slender film [17]. Figure 2 shows the schematic representation of the flow showing a film flowing vertically down, subjected to an electric field imposed across electrodes separated by a distance d.Figure 1
Flame front position.Figure 2
Schematic representation of the flow showing a film flowing vertically down.


## 2. Lie Symmetry of Generalized Modified Kuramoto-Sivashinsky Equation

In this part, we compute our main results.Consider one parameter local Lie group of transformation for the independent factorsx, t and dependent factor u as follows:
(3)x∗=x+δαx,t,u+Oδ2,t∗=t+δβx,t,u+Oδ2,u∗=u+δγx,t,u+Oδ2,in which δ∈ℝ is the parameter.Proposition 1.
For alln≥1,n∈N, the algebra of symmetries of
(4)unut+uxx+uxxxx+λ−1ux2−σuxx2=0,is 2-dimensional Abelian Lie algebra.Proof.
The general infinitesimal generator (symmetries) is(5)H=αx,t,u∂x+βx,t,u∂t+γx,t,u∂u.
The derivation ofnth prolongation of H(6)prnH=H+∑i=1q∑PγiPx,un∂∂uPi,interprets the relating jet space Qn⊂X×Un, where q is a dependent variables, and P=P1,P2,⋯Pk with 1≤Pk≤p, 1≤k≤n and
(7)γiPx,un=DPγi−∑l=1pαluil+∑l=1pαluiP,l,where p is an independent variable and uil=∂ui/∂ul and uiP,l=∂uiP/∂xl.
The fourth-order prolongation ofH is
(8)Pr4H=H+γx∂∂ux+γt∂∂ut+γxx∂∂uxx+γxt∂∂uxt+γtt∂∂utt+γxxx∂∂uxxx+γxxt∂∂uxxt+γxtt∂∂uxtt+γttt∂∂uttt+γxxxx∂∂uxxxx+γxxxt∂∂uxxxt+γxxtt∂∂uxxtt+γxttt∂∂uxttt+γtttt∂∂utttt,(9)pr4unut+uxx+uxxxx+λ−1ux2−σuxx2≡0mod2,nun−1γt+γxx+γxxxx+2λ−1uxγx−2σuxxγxx≡0mod2.
We can calculateγt,γx, γxx, and γxxxx from equation (7) such that
(10)γt=Dtγ−αux−βut+αuxt+βutt,γx=Dxγ−αux−βut+αuxx+βuxt,γxx=Dx2γ−αux−βut+αuxxx+βuxxt,γxxxx=Dx4γ−αux−βut+αuxxxxx+βuxxxxt,where Dx and Dt are total derivatives.☐Putting all the above in equation (9) and eliminating ut by using the relation ut=1/unσuxx2−λ−1ux2−uxx−uxxxx, we get a polynomial equation containing the different differentials of u. Equating the coefficient of u to zero, which are some derivatives of α, β, and γ, it gives the total set of determining equations.
(11)αx=αt=αu=0,βx=βt=βu=0,γ=0.This gives(12)αx,t,u=0,βx,t,u=0,γx,t,u=0.This implies that the Lie group (algebra) of infinitesimal generators of equaution (2) is comprised of two vector fields:
(13)H1=∂x,H2=∂t.The commutator table of the Lie group for equation (2) is given as in Table 1,Table 1
Commutator table.
..,..H1H2H100H200The adjoint table of infinitesimal symmetries for equation (2) is given as in Table 2,Table 2
Adjoint table.
adH1H2H1H1H2H2H1H2In this case, we have only two different basis for a Lie algebra of symmetries.Hence, this shows that the group of symmetries of equation (2) is two dimensional and tables ensure that it is abelian.Proposition 2.
For alln>1 and n∈N, the group of symmetries of
(14)eunut+uxx+uxxxx+λ−1ux2−σuxx2=0,is two-dimensional abelian.Proof.
The infinitesimal generator is(15)X=αx,t,u∂x+βx,t,u∂t+γx,t,u∂u.
In order to find the symmetry group of equation (14), we have to apply invariance condition that is X45≡0mod6 on equation .(14) where X4 is the fourth-order prolongation of X given as
(16)Pr4X=X+γx∂∂ux+γt∂∂ut+γxx∂∂uxx+γxt∂∂uxt+γtt∂∂utt+γxxx∂∂uxxx+γxxt∂∂uxxt+γxtt∂∂uxtt+γttt∂∂uttt+γxxxx∂∂uxxxx+γxxxt∂∂uxxxt+γxxtt∂∂uxxtt+γxttt∂∂uxttt+γtttt∂∂utttt.
After applying an invariance condition on equation (14), we get
(17)nun−1eunγt+γxx+γxxxx+2λ−1uxγx−2σuxxγxx≡0.mod2.
We can calculateγt, γx, γxx, and γxxxx from equation (7) such that
(18)γt=Dtγ−αux−βut+αuxt+βutt,γx=Dxγ−αux−βut+αuxx+βuxt,γxx=Dx2γ−αux−βut+αuxxx+βuxxt,γxxxx=Dx4γ−αux−βut+αuxxxxx+βuxxxxt,where Dx and Dt are total derivatives.☐After putting all the above in equation (17), and eliminating ut by using the relation ut=1/eunσuxx2−λ−1ux2−uxx−uxxxx, we get a polynomial equation containing the different differentials of u. Equating the coefficient of u to zero, which are some derivatives of α, β, and γ, it gives the total set of determining equations, as given byαx=αt=αu=0, βx=βt=βu=0, γ=0,that gives(19)αx,t,u=0,βx,t,u=0,γx,t,u=0.This implies that the Lie group (algebra) of infinitesimal generators of equation (9) comprises two vector fields. Following, Table 3 gives the commutator table as
(20)X1=∂x,X2=∂t.Table 3
Commutator table.
..,..X1X2X100X200The commutator table of the Lie group for equation (14) is given in Table 3,The adjoint table of infinitesimal symmetries for equation (14) is given in Table 4,Table 4
Adjoint table.
adX1X2X1X1X2X2X1X2In this case, we have only two different basis for Lie algebra.Hence, this shows that the group of symmetries of equation (14) is two-dimensional abelian.


### 2.1. Symmetry Algebra foreunut+uxx+uxxxx+λ−1ux2−σuxx2=0 When n=1

Forn=1, the equation is
(21)Δ:euut+uxx+uxxxx+λ−1ux2−σuxx2=0.The general infinitesimal generator is(22)V=τx,t,u∂x+μx,t,u∂t+νx,t,u∂u.In order to find the symmetry algebra, we have to apply invariance condition that is(23)VnΔ≡0modΔ,on equation (21) where V4 is the fourth-order prolongation of V such that
(24)Pr4V=V+νx∂∂ux+νt∂∂ut+νxx∂∂uxx+νxt∂∂uxt+νtt∂∂utt+νxxx∂∂uxxx+νxxt∂∂uxxt+νxtt∂∂uxtt+νttt∂∂uttt+νxxxx∂∂uxxxx+νxxxt∂∂uxxxt+νxxtt∂∂uxxtt+νxttt∂∂uxttt+νtttt∂∂utttt.After applying invariance condition (23) on equation (21)
(25)ueuνt+νxx+νxxxx+2λ−1uxνx−2σuxxνxx≡0modΔ,where νt,νx, νxx, and νxxxx from equation (7) such that
(26)νt=Dtν−τux−μut+τuxt+μutt,νx=Dxν−τux−μut+τuxx+μuxt,νxx=Dx2ν−τux−μut+τuxxx+μuxxt,νxxxx=Dx4ν−τux−μut+τuxxxxx+μuxxxxt,where Dx and Dt are total derivative.Putting all the above in equation (25), we eliminateut by using the relation ut=1/euσuxx2−λ−1ux2−uxx−uxxxx and get a polynomial equation containing the different differentials of u. Equating the coefficient of u to zero, which are some derivatives of τ, μ, and ν, it gives the total set of determining equations.τx=τt=τu=0, μx=μt=μu=0, ν=0,that gives(27)τx,t,u=c3,μx,t,u=c1t+c2,νx,t,u=c1.This implies that the Lie group (algebra) of infinitesimal generators of equation (21) is comprised of three vector fields:
(28)V1=∂u+t∂t,(29)V2=∂t,(30)V3=∂x.The commutator table of the Lie group for equation (21) is given in Table 5,Table 5
Commutator table.
..,..V1V2V3V10−V20V2V200V3000The adjoint table of infinitesimal symmetries for equation (21) is given in Table 6,Table 6
Adjoint table.
adV1V2V3V1V1V2eεV3V2V1−εV2V2V3V3V1V2V3In this case, we have three different Lie algebras.Theorem 3.
The algebra of symmetries of Kuramoto-Sivashinsky type equation is(31)eunut+uxx+uxxxx+λ−1ux2−σuxx2=0,where it is two-dimensional abelian for all n>1, nεN and three-dimensional nonabelian for n=1.Proof.
The proof follows easily using Propositions1 and 2.☐Theorem 4.
IfGsix,t,u be the one parameter group generated by equation (28) then
(32)Gs1x,t,u=x,est,u,Gs2x,t,u=x,t+s,u,Gs3x,t,u=x+s,t,u.
There will be a family of solutions to each one parameter subgroups of the full symmetry group of a system called group invariant solutions.Theorem 5.
Ifu=fx,t is a solution of equation (21), so are the functions
(33)u~1=fx,e−st,u~2=fx,t−s,u~3=fx−s,t,where u~i=Gsi∗fx,t, i=1,2,3 and s<<1 is any positive number.Proof.
The one parameter Lie group of equation (21) is
(34)Gs1:x,t,u⟶x,est,u,with the infinitesimal generator
(35)V1=t∂t+∂u,if u~1x,t is any function then it transformed by Gs1 as
(36)u~1=u,u~1=fx,t,now
(37)x~,t~=x,e−st,therefore,
(38)u~1=fx,e−st.
The graph foru~1=fx,e−st is given in Figure 3.
The one parameter Lie group of equation (21) is
(39)Gs2:x,t,u⟶x,t−s,u,with the infinitesimal generator
(40)V2=t∂t,if u~2x,t is any function then it transformed by Gs2 as
(41)u~2=u,u~2=fx,t,now
(42)x~,t~=x,t−s,therefore
(43)u~2=fx,t−s.
The graph foru~2=fx,t−s is given in Figure 4.
The one parameter Lie group of equation (21) is
(44)Gs3:x,t,u⟶x−s,t,u,with the infinitesimal generator
(45)V3=t∂x,if u~3x,t is any function then it transformed by Gs3 as
(46)u~3=u,u~3=fx,t,now
(47)x~,t~=x−s,t,therefore
(48)u~3=fx−s,t.
The graph foru~3=fx−s,t as a solution is given in Figure 5.☐Figure 3
Foru1x,t=sinx+e−st,s=0.00001.Figure 4
Foru1x,t=logx+t−s,s=0.00001.Figure 5
Foru3x,t=x−s+cost,s=0.00001.


## 2.1. Symmetry Algebra foreunut+uxx+uxxxx+λ−1ux2−σuxx2=0 When n=1

Forn=1, the equation is
(21)Δ:euut+uxx+uxxxx+λ−1ux2−σuxx2=0.The general infinitesimal generator is(22)V=τx,t,u∂x+μx,t,u∂t+νx,t,u∂u.In order to find the symmetry algebra, we have to apply invariance condition that is(23)VnΔ≡0modΔ,on equation (21) where V4 is the fourth-order prolongation of V such that
(24)Pr4V=V+νx∂∂ux+νt∂∂ut+νxx∂∂uxx+νxt∂∂uxt+νtt∂∂utt+νxxx∂∂uxxx+νxxt∂∂uxxt+νxtt∂∂uxtt+νttt∂∂uttt+νxxxx∂∂uxxxx+νxxxt∂∂uxxxt+νxxtt∂∂uxxtt+νxttt∂∂uxttt+νtttt∂∂utttt.After applying invariance condition (23) on equation (21)
(25)ueuνt+νxx+νxxxx+2λ−1uxνx−2σuxxνxx≡0modΔ,where νt,νx, νxx, and νxxxx from equation (7) such that
(26)νt=Dtν−τux−μut+τuxt+μutt,νx=Dxν−τux−μut+τuxx+μuxt,νxx=Dx2ν−τux−μut+τuxxx+μuxxt,νxxxx=Dx4ν−τux−μut+τuxxxxx+μuxxxxt,where Dx and Dt are total derivative.Putting all the above in equation (25), we eliminateut by using the relation ut=1/euσuxx2−λ−1ux2−uxx−uxxxx and get a polynomial equation containing the different differentials of u. Equating the coefficient of u to zero, which are some derivatives of τ, μ, and ν, it gives the total set of determining equations.τx=τt=τu=0, μx=μt=μu=0, ν=0,that gives(27)τx,t,u=c3,μx,t,u=c1t+c2,νx,t,u=c1.This implies that the Lie group (algebra) of infinitesimal generators of equation (21) is comprised of three vector fields:
(28)V1=∂u+t∂t,(29)V2=∂t,(30)V3=∂x.The commutator table of the Lie group for equation (21) is given in Table 5,Table 5
Commutator table.
..,..V1V2V3V10−V20V2V200V3000The adjoint table of infinitesimal symmetries for equation (21) is given in Table 6,Table 6
Adjoint table.
adV1V2V3V1V1V2eεV3V2V1−εV2V2V3V3V1V2V3In this case, we have three different Lie algebras.Theorem 3.
The algebra of symmetries of Kuramoto-Sivashinsky type equation is(31)eunut+uxx+uxxxx+λ−1ux2−σuxx2=0,where it is two-dimensional abelian for all n>1, nεN and three-dimensional nonabelian for n=1.Proof.
The proof follows easily using Propositions1 and 2.☐Theorem 4.
IfGsix,t,u be the one parameter group generated by equation (28) then
(32)Gs1x,t,u=x,est,u,Gs2x,t,u=x,t+s,u,Gs3x,t,u=x+s,t,u.
There will be a family of solutions to each one parameter subgroups of the full symmetry group of a system called group invariant solutions.Theorem 5.
Ifu=fx,t is a solution of equation (21), so are the functions
(33)u~1=fx,e−st,u~2=fx,t−s,u~3=fx−s,t,where u~i=Gsi∗fx,t, i=1,2,3 and s<<1 is any positive number.Proof.
The one parameter Lie group of equation (21) is
(34)Gs1:x,t,u⟶x,est,u,with the infinitesimal generator
(35)V1=t∂t+∂u,if u~1x,t is any function then it transformed by Gs1 as
(36)u~1=u,u~1=fx,t,now
(37)x~,t~=x,e−st,therefore,
(38)u~1=fx,e−st.
The graph foru~1=fx,e−st is given in Figure 3.
The one parameter Lie group of equation (21) is
(39)Gs2:x,t,u⟶x,t−s,u,with the infinitesimal generator
(40)V2=t∂t,if u~2x,t is any function then it transformed by Gs2 as
(41)u~2=u,u~2=fx,t,now
(42)x~,t~=x,t−s,therefore
(43)u~2=fx,t−s.
The graph foru~2=fx,t−s is given in Figure 4.
The one parameter Lie group of equation (21) is
(44)Gs3:x,t,u⟶x−s,t,u,with the infinitesimal generator
(45)V3=t∂x,if u~3x,t is any function then it transformed by Gs3 as
(46)u~3=u,u~3=fx,t,now
(47)x~,t~=x−s,t,therefore
(48)u~3=fx−s,t.
The graph foru~3=fx−s,t as a solution is given in Figure 5.☐Figure 3
Foru1x,t=sinx+e−st,s=0.00001.Figure 4
Foru1x,t=logx+t−s,s=0.00001.Figure 5
Foru3x,t=x−s+cost,s=0.00001.


## 3. Optimal System of Subalgebras

This is remarkable that the Lie symmetry technique assumes a significant part to determine the solutions of PDEs as well as performing the symmetric reductions. Every combination (should be linear) of infinitesimal symmetries(generators) is a result of another infinitesimal symmetry(generator). As any transformation in the full symmetry groups plot a solution to another, it is sufficient to determine the invariant solution which are not related by transformations in the full symmetry group; this prompted the Optimal system [18, 19].Theorem 6.
A 1D optimal system of equation (21) is given by those generated by
(49)Y1=V1Y2=V2,Y3=V3,Y4=V1+V3,Y5=V3−V1.Proof.
Since the combination of vector field (infinitesimal generator) is also a vector field. Consider a linear combinationV of V1, V2, and V3,
(50)V=∑i=13=biVi,
a nonzero vector field. Here, for proof, we will improve as many of the coefficientbi′s as possible by using adjoint application on V.☐Case 1.
Firstly assume thatb3≠0 then
(51)V=b1V1+b2V2+V3,acting on V with Adjexpb2/b1V2 by using the adjoint table (adjoint Table 3)
(52)V′=Adjexpb2b1V2V=b1V1+V3.
Whenb1>0, then we get Y4.
Whenb1<0, then we get Y5.
Whenb1=0, then we get Y3.Case 2.
Letb3=0 and b1=0,
(53)V=b2V2,when b1=1 then we get Y2, Let b3=0, b2=0 and b1, then we get Y1 There is no any more cases for consultation and the proof is complete.


## 4. Lie Invariants and Similarity Solutions

We can discover that the invariants correlate with the infinitesimal symmetries (28); they can be determined by solving the equations (by using characteristic method). For V2=∂t, the characteristic equation is dx/0=dt/1=du/0 and the corresponding invariants of this system x=r and u=w.We obtain a similar solution of the formw=wr, and we put it into equation (21) to obtain the form of the function w, and then, we conclude that w=wr=wx solution of the following differential equation as similarity reduce equation:
(54)wrr+wrrrr+λ−1wr2−σwrr2=0.For other example, take V3=∂x; the characteristic equation for this has the form dx/1=dt/0=du/0 so the corresponding invariants are t=r and u=w.Taking into account the last invariants, the following similarity solution is obtainedw=wr=wt where the solution satisfied the similarity reduce equation:
(55)ewwr=0.


## 5. Conclusions

The present paper addresses Lie symmetries for some general cases of modified one-dimensional Kuramoto-Sivashinky equation (MKS) as well as its similarity solutions using a symmetry operator. In Section2, we discussed general results for Lie algebras for some general cases of MKS and provide a comparison between them and obtained some general results. In Section 3, we find the optimal system for (MKS). In the last section, we obtained similarity solutions and Lie invariants.Remarks.
It is worthmentioning thatfu can be any arbitrary function. For other similar functions chosen as fu, the procedure for symmetry analysis can be very tedious and symmetry algebra can be different.



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