A multiple exp-function method for nonlinear differential equations and its application

TL;DR

This paper introduces a multiple exp-function method for solving nonlinear partial differential equations, enabling systematic derivation of multi-wave solutions including solitons, with implementation demonstrated using Maple on a high-dimensional equation.

Contribution

It presents a new systematic method that generalizes Hirota's scheme for obtaining explicit multi-wave solutions of nonlinear PDEs, compatible with computer algebra systems.

Findings

Successfully derived explicit 1-, 2-, and 3-wave solutions including solitons.

Implemented the method using Maple for a 3+1 dimensional equation.

Generated visual plots for specific parameter cases.

Abstract

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.