Lie Symmetry Analysis and Similarity Solutions for the Jimbo-Miwa Equation and Generalisations

TL;DR

This paper applies Lie symmetry analysis to the Jimbo-Miwa equation and its extensions, deriving new reductions, solutions, and integrability insights, with some reductions leading to hypergeometric and linearisable equations.

Contribution

It provides new similarity reductions and solutions for the Jimbo-Miwa equation and its extensions, including hypergeometric solutions and integrability analysis using Lie groups.

Findings

Traveling-wave solutions are similar across all equations.

New reductions lead to hypergeometric and linearisable equations.

Singularity analysis confirms integrability of certain reduced equations.

Abstract

We study the Jimbo-Miwa equation and two of its extended forms, as proposed by Wazwaz et al, using Lie's group approach. Interestingly, the travelling-wave solutions for all the three equations are similar. Moreover, we obtain certain new reductions which are completely different for each of the three equations. For example, for one of the extended forms of the Jimbo-Miwa equation, the subsequent reductions leads to a second-order equation with Hypergeometric solutions. In certain reductions, we obtain simpler first-order and linearisable second-order equations, which helps us to construct the analytic solution as a closed-form function. The variation in the nonzero Lie brackets for each of the different forms of the Jimbo-Miwa also presents a different perspective. Finally, singularity analysis is applied in order to determine the integrability of the reduced equations and of the different forms of the Jimbo-Miwa equation.