Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

Abstract

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

“…Such a procedure allows the further application of the SEsM. In this direction, the SEsM covers all previous methodologies for finding exact solutions of PDEs to this end, as proved in [ 73 , 74 , 75 , 76 , 77 , 78 ]. Although the SEsM is a relatively new methodology, its application for finding exact solutions of different NPDEs can be seen in [ 79 , 80 , 81 , 82 , 83 ].…”

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

“…Kudryashov [ 67 , 68 , 69 , 70 ] also used various transformations in order to transform the non-linearity of a generalized evolution equation of the wave dynamics and to obtain exact solutions of this equation. More results connected to the application of the MSE can be seen, for example, in [ 71 , 72 , 73 , 74 , 75 , 76 , 77 ].…”

“…Some elements of SEsM [ 77 , 78 , 79 , 80 , 81 ] can be seen in our publications written years ago [ 82 , 83 , 84 , 85 , 86 , 87 , 88 ]. About 10 years ago [ 89 , 90 ], we used the ordinary differential equation of Bernoulli as the simplest equation [ 91 ] and applied a methodology called the Modified Method of Simplest Equation (MMSE) to model non-linear partial differential equations from ecology and population dynamics [ 92 ].…”

“…Over the course of the years, MMSE was extended to SEsM [ 77 ]. SEsM is connected to the possibility of using more than one simple equation in order to construct the solution of the solved non-linear differential equation.…”

“…A realization of this possibility was shown in [ 104 ]. The first description of SEsM was made in [ 78 ] and then in [ 77 , 79 , 80 , 81 , 105 ]. Several examples of the methodology and its applications are presented in [ 106 , 107 , 108 , 109 , 110 , 111 ].…”