Exact soliton solutions of nonlinear partial differential equations by the simplest and the extended simplest equation method

Abstract

In this paper six different exact solution methods have been presented and applied to solve Korteweg and de Vries (KdV) and Burgers equations whose solutions are proportional to the nonlinear term coefficients The new exact solutions of KdV and Burgers equations with the nonlinear term coefficients being arbitrary constants are derived by the simplest equation method (SEM) with Bernoulli equation as the simplest equation It is shown that the proposed exact solutions overcome the long existing problem of discontinuity and can be successfully reduced to linearity when the nonlinear term coefficients approach to zero Comparison of the existing and new soliton solutions is presented A new phenomenon named soliton sliding is observed Moreover we extend the SEM by choosing Burgers equation as the simplest equation The reason of setting Burgers equation as the simplest equation is due it being completely integrable equation The extended SEM is applied to handle two completely integrable equations KdV and the potential KdV equations The general forms of the multi-soliton solutions are formally established Unlike Hirota’s method the results confirm the extended SEM is concise and effective for constructing multi-soliton solutions Accordingly we believe that solitary solutions and multi-soliton solutions existing for other classes of nonlinear mathematic physics models can be easily solved by the SEM and the extended SEM Further work on these aspects is recommended

Date of Award	2015 Oct 29
Original language	English
Supervisor	Sen-Yung Lee (Supervisor)