A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation

TL;DR

This paper introduces a rational function transformation method for solving nonlinear PDEs, unifying several existing techniques, and applies it to find exact solutions of the 3+1 dimensional Jimbo-Miwa equation.

Contribution

The paper presents a systematic rational function transformation approach that unifies multiple solution methods for nonlinear PDEs, demonstrated on the Jimbo-Miwa equation.

Findings

Derived exact solutions to the 3+1 dimensional Jimbo-Miwa equation.

Unified various solution techniques into a single rational function framework.

Constructed a Bäcklund transformation for the equation.

Abstract

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a B\"acklund transformation.