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

Lie symmetry analysis of di ﬀ erential equations proves to be a powerful tool to solve or at least reduce the order and nonlinearity of the equation. Symmetries of di ﬀ erential equations is the most signi ﬁ cant 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 ﬁ nd symmetries of some general classes of KS-type equation. We also compute transformed equivalent equations and some invariant solutions of this equation.


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][2][3][4][5][6][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 onedimensional 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 onedimensional 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 onedimensional modified KS-type equation is 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 We seek the Lie symmetry algebras for this GMKS equations for f ðuÞ = u n , f ðuÞ = e nu , and f ðuÞ = e u n 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.

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 factors x, t and dependent factor u as follows: in which δ ∈ ℝ is the parameter.

Proposition 1.
For all n ≥ 1, n ∈ N, the algebra of symmetries of is 2-dimensional Abelian Lie algebra.

Proof. The general infinitesimal generator (symmetries) is
The derivation of nth prolongation of H interprets the relating jet space Q n ⊂ X × U n , where q is a dependent variables, and P = ðP 1 , P 2 , ⋯P k Þ with 1 ≤ P k ≤ p, 1 ≤ k ≤ n and where p is an independent variable and u i l = ∂u i /∂u l and u i P,l = ∂u i P /∂x l . The fourth-order prolongation of H is

Journal of Function Spaces
We can calculate γ t ,γ x , γ xx , and γ xxxx from equation (7) such that where D x and D t are total derivatives.☐ Putting all the above in equation (9) and eliminating u t by using the relation u t = 1/u n ðσu 2 xx − ðλ − 1Þu 2 x − u xx − u xxxx Þ, 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.
This gives This implies that the Lie group (algebra) of infinitesimal generators of equaution (2) is comprised of two vector fields: The commutator table of the Lie group for equation (2) is given as in Table 1, The adjoint table of infinitesimal symmetries for equation (2) is given as in Table 2, In 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 all n > 1 and n ∈ N, the group of symmetries of is two-dimensional abelian.
Proof. The infinitesimal generator is In order to find the symmetry group of equation (14), we have to apply invariance condition that is X ð4Þ ð5Þ ≡ 0 mod ð6Þ on equation . (14) where X ð4Þ is the fourth-order prolongation of X given as After applying an invariance condition on equation (14), we get We can calculate γ t , γ x , γ xx , and γ xxxx from equation (7) such that where D x and D t are total derivatives.☐

Journal of Function Spaces
After putting all the above in equation (17), and eliminating u t by using the relation u t = 1/e u n ðσu 2 xx − ðλ − 1Þu 2 x − u xx − u xxxx Þ, 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 by 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 The 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, In 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.

Symmetry Algebra for e u n u t + u xx
The general infinitesimal generator is In order to find the symmetry algebra, we have to apply invariance condition that is on equation (21) where V ð4Þ is the fourth-order prolongation of V such that After applying invariance condition (23) on equation (21) where ν t ,ν x , ν xx , and ν xxxx from equation (7) such that where D x and D t are total derivative. Putting all the above in equation (25), we eliminateu t by using the relation u t = 1/e u ðσu 2 xx − ðλ − 1Þu 2 x − u xx − u xxxx Þ 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.
This implies that the Lie group (algebra) of infinitesimal generators of equation (21) is comprised of three vector fields: The commutator table of the Lie group for equation (21) is given in Table 5, The adjoint table of infinitesimal symmetries for equation (21) is given in Table 6, In this case, we have three different Lie algebras. Table 3: Commutator table. ::, ::

Journal of Function Spaces
where it is two-dimensional abelian for all n > 1, nεN and three-dimensional nonabelian for n = 1: Proof. The proof follows easily using Propositions 1 and 2.☐ Theorem 4. If G i s ðx, t, uÞ be the one parameter group generated by equation (28) then There will be a family of solutions to each one parameter subgroups of the full symmetry group of a system called group invariant solutions.
Proof. The one parameter Lie group of equation (21) is with the infinitesimal generator ifũ 1 ðx, tÞ is any function then it transformed by G 1 s as therefore,ũ The graph forũ 1 = f ðx, e −s tÞ is given in Figure 3. The one parameter Lie group of equation (21) is with the infinitesimal generator ifũ 2 ðx, tÞ is any function then it transformed by G 2 s as The graph forũ 2 = f ðx, t − sÞ is given in Figure 4. The one parameter Lie group of equation (21) is ::, :: ifũ 3 ðx, tÞ is any function then it transformed by G 3 s as thereforeũ The graph forũ 3 = f ðx − s, tÞ as a solution is given in Figure 5.☐

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 Proof. Since the combination of vector field (infinitesimal generator) is also a vector field. Consider a linear combination V of V 1 , V 2 , and V 3 , a nonzero vector field. Here, for proof, we will improve as many of the coefficient b i ′ s as possible by using adjoint application on V.☐ Case 1. Firstly assume that b 3 ≠ 0 then acting on V with Adjðexp ðb 2 /b 1 ÞV 2 Þ by using the adjoint table (adjoint Table 3) When b 1 > 0, then we get Y 4 . When b 1 < 0, then we get Y 5 . When b 1 = 0, then we get Y 3 .
when b 1 = 1 then we get Y 2 , Let b 3 = 0, b 2 = 0 and b 1 , then we get Y 1 There is no any more cases for consultation and the proof is complete.

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 V 2 = ∂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 form w = wðrÞ, and we put it into equation (21) to obtain the form of the function w, and then, we conclude that w = wðrÞ = wðxÞ solution of the following differential equation as similarity reduce equation: For other example, take V 3 = ∂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 obtained w = wðrÞ = wðtÞ where the solution satisfied the similarity reduce equation: e w w r = 0:

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 Section 2, 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 that f ðuÞ can be any arbitrary function. For other similar functions chosen as f ðuÞ, the procedure for symmetry analysis can be very tedious and symmetry algebra can be different.

Data Availability
Data sharing is not applicable to this article as no dataset was generated or analyzed during the current study

Conflicts of Interest
The authors declare that they have no conflicts of interest.