Multiple Soliton Solutions of the Sawada-Kotera Equation with a Nonvanishing Boundary Condition and the Perturbed Korteweg de Vries Equation by Using the Multiple Exp-Function Scheme

1Department of Mathematical Sciences, North-West University, Private Bag X 2046, Mmabatho 2735, South Africa 2Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar 44891-63157, Iran 3School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, China 4Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran


Introduction
In the applied science, NLEEs are extensively used in theoretical studies to model a wide range of nonlinear phenomena.To comprehend the mechanisms of nonlinear phenomena, it is vital to investigate the solutions of NLEEs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].One specific tool that has recently achieved a special interest from academic researchers is the multiple exp-function scheme [16,17].The MEFS supposes that the multisoliton solutions of NLEEs can be presented as (, ) = / in which  and  are polynomials of exponential functions.The SK equation with a nonvanishing boundary condition [18,19]   +  (  5  + 3 2 +   ) is one of NLEEs that models the evolution of steeper waves of shorter wavelength than those explained by the KdV equation and its perturbed form.By using the binary-Bell-polynomial Hirota method and symbolic computation, the bilinear form and N-soliton solutions for this model were derived in [19].
The perturbed form of KdV equation [18,[20][21][22]] is another kind of NLEEs describing some arrays of wave crests.The bilinear form, Bäcklund transformation, superposition formulae, and N-soliton solutions in terms of the Wronskian were done in [22].For computational purposes, we use the transformation Advances in Mathematical Physics to convert (1) and (2) into the following, respectively: and The key goal of present work is applying the MEFS to generate the multiple soliton solutions of the models (1) and (2).

Multiple Exp-Function Method
The key steps of MEFS can be summarized as follows [16,17].
Step .Let us consider the following (1 + 1)-dimensional NLEE: Step .Suppose the solution of above NLEE can be expressed as in which  , and  , are unknowns to be determined and Step .Substituting (7) and its derivatives into (6) yields the following transformed equation: Step .By setting the numerator of the function (, ,  1 ,  2 , ⋅ ⋅ ⋅ ,   ) to zero, we will reach an algebraic system which its solution yields the multiple wave solution of (6) as . .Multiple Soliton Solutions of SK Equation with a Nonvanishing Boundary Condition.In the current subsection, the multiple soliton solutions of SK equation with a nonvanishing boundary condition are derived through the MEFS.
. . .One-Soliton Solution of ( ).To obtain one-soliton solution, it is assumed where  1 is a constant and the dispersion is Now, applying the MEFS results in one-soliton solution can be presented as where  1 is arbitrary but  1 is defined by (13).

. . . Two-Soliton Solution of ( ).
To seek two-soliton solution, the following ansatz is considered in which  and  are defined as Now, by applying the MEFS, we acquire . . .ree-Soliton Solution of ( ).To derive three-soliton solution, it is assumed in which  and  are defined as Now, applying the MEFS yields .

. Multiple Soliton Solutions of pKdV Equation ( ).
In the present subsection, the multiple soliton solutions of pKdV equation are obtained through the MEFS.
. . .One-Soliton Solution of ( ).To obtain one-soliton solution, it is assumed where  1 is a constant and and the dispersion relation is Now, applying the MEFS results in and, so, the resulting one-soliton solution can be presented as where  1 is arbitrary but  1 is defined by (26).

. . . Two-Soliton Solution of ( ).
To seek two-soliton solution, the following ansatz is considered in which  and  are defined as Now, by applying the MEFS, we acquire A profile of the evolution of the two-soliton solution (29) is given in Figure 1.
. . .ree-Soliton Solution of ( ).To derive three-soliton solution, it is assumed in which  and  are defined as Now, applying the MEFS yields (36) A profile of the evolution of the three-soliton solution (33) is given in Figure 2.

Concluding Remarks
We note that multiple soliton solutions of (1) and ( 2) are in agreement with [19,21,22].Moreover, we should emphasize the approach employed here was independent of the bilinear forms, simplified Hirota method, Darboux transformation method, or Bell polynomial technique.Finally we can say the multiple exp-function algorithm is an elegant and versatile method that can be adopted to other NLEEs of mathematical physics.