Group classification and exact solutions of a class of nonlinear waves

TL;DR

This paper extends a group classification method to nonlinear wave equations, identifies a specific model, and derives exact solutions including multi-solitons using symmetry and Hirota methods.

Contribution

It introduces the method of indeterminates for group classification and applies it to a family of nonlinear wave equations, finding exact solutions and analyzing symmetry actions.

Findings

Successfully classified a family of nonlinear wave equations

Derived exact traveling wave and multi-soliton solutions

Provided insights into the symmetry structure of the equations

Abstract

We apply an extension of a new method of group classification to a family of nonlinear wave equations labelled by two arbitrary functions, each depending on its own argument. The results obtained confirm the efficiency of the proposed method for group classification, termed the method of indeterminates. A model equation from the classified family of fourth order Lagrange equations is singled out. Travelling wave solutions of the latter are found through a similarity reduction by variational symmetry operators, followed by a double order reduction into a second order ordinary differential equation. Multi-soliton solutions and other exact solutions are also found by various methods including Lie group and Hirota methods. The most general action of the full symmetry group on any given solution is provided. Some remarkable facts on Lagrange equations emerging from the whole study are outlined.