Lie Symmetry Solutions of Sawada-Kotera Equation

In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth-order Sawada-Kotera equation of the form . The technique is among the most powerful approaches currently used to achieve precise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generators, and invariant transformations/solutions hence symmetry solutions of the fifth-order Sawada-Kotera equation.

The technique is among the most Introduction Recently, many phenomena in physical, engineering and mathematical fields are expressed in terms of Partial Differential Equations and Ordinary Differential Equations. Among the many approaches that have been recommended for obtaining the precise solutions of the given differential equations is Lie symmetry analysis, which provides a very effective procedure.
The other methods include Jacobi elliptic function expansion technique as discussed by Liu et al. [1], the Hirota's bilinear transformation scheme as demonstrated by Hirota [2], Backlund transformation technique as illustrated by Rogers and Shadwick [3], homotopy analysis technique as discussed by Liao [4], variational iteration scheme as demonstrated by Noor and Mohyud [5].
Recently, Lie symmetry analysis method is effectively used in the study of several differential equations to obtain solutions such as Burgers equation as discussed by Oduor [6], nonlinear evolution equations having advanced order nonlinearity as discussed by Khongorzul et al. [7], nonlinear beam equation as deliberated by [8,9], nonlinear ordinary differential equation as discussed by Aminer [10], a scheme of nonlinear partial differential equation as discussed by Andronikos et al. [11], nonlinear Helmholtz equation as illustrated by Sakkaravarthi [12], reactiondiffusion equation as demonstrated by Yildirim and Pinar [13] and many more.
In this article, we apply Lie symmetry analysis method to develop prolongations, symmetry groups, invariant solutions, exact solutions and symmetry solutions of Sawada-Kotera equation of the form u u x t = 2 The outline of this article is as follows. In section 2, we present the methods of using Lie symmetry analysis. Section 3 presents symmetry solutions of equation (1.1). Finally, section 4 gives the conclusion and remarks.

Lie symmetry analysis
We let the generator G of (1.1) to be of the form All the coefficient functions u U x t u  = for a symmetry group of equation (1.1) For the symmetry state to be satisfied by (1.1), we then have We then equate the coefficients of several polynomials in the first to the fifth partial derivatives of u . This is satisfied when t u is replaced by whenever it occurs in the equation hence obtaining the determining equations for obtaining the Lie groups of (1.1) The solutions from the determining equations become Thus the infinitesimal symmetries/generators of (1.1) are as given below Lie groups admitted by infinitesimal generators are obtained by solving the corresponding Lie equations through exponentiation. This leads to Or its system of characteristics Here we consider the group transformations that arise from all the generators of (1.1) We then allocate one of the invariants as a function of the other as given below We then substitute for  , in (2.1) to get an ordinary differential equation for the function ( ) k  of one variable. By doing this, we decrease the figure of independent variables by one.
We now show the list of generators ( i X ) and their equivalent Invariant Solutions (u)

Case 1
For the infinitesimal generator, 1 When we substitute it into equation (1.1) we get the trivial solution to be

Case 2
For the generator 2 w t When it is substituted into equation (1.1), the equation is reduced into the following ordinary differential equation For the generator, 3 52 Substituting into equation (1.1), will reduce the equation into the following ordinary differential equation The exact solutions of some ODEs or some PDEs that are lower in order than the original PDE can be obtained as discussed by Galaktionov and Svirshchevskii [14]. Thus in our case, the exact solutions of the fifth-order Sawada-Kotera equation are found from the obtained ODE's. Even though there are so many methods to be applied in solving differential equations such as reduction by quadratures, and so many others but it is not always easy for simple semilinear PDE's.
Therefore, the analytic solutions of the reduced equations are found through the power series technique since this scheme is used in solving all kinds of differential equations including those with non-constant coefficients, as discussed elsewhere in [15][16][17][18]. Thus the exact analytic solutions obtained are the solutions of the original PDE.
In view of (2.3) we have Setting the a=0 we get  Whenever a known solution of (1.1) is given as ( ) , u u x t =

Solution 3
Substituting the exact solution

Conclusion and Remarks
In this article, the symmetry solutions of the fifth-order Sawada-Kotera equation have been acquired by means of Lie symmetry analysis method. Also, deliberation on the group-invariant solutions and the exact analytic solutions to the equation through the power series technique has been done. Our obtained symmetry solutions demonstrate that Lie symmetry analysis technique is a candid and best mathematical tool to obtain analytical results of highly nonlinear PDE's. Furthermore, this technique can be more efficiently used to study other nonlinear PDE's, which are often used in mathematical fields, engineering fields, and physical sciences.

Conflicts of Interest
There are no conflicts of interest

Funding Statement
The research was self-sponsored by the author.