Lie symmetries and similarity solutions for a family of 1+1 fifth-order partial differential equations

TL;DR

This paper classifies Lie symmetries of a family of 1+1 fifth-order PDEs related to kink solutions, deriving similarity solutions including travel-wave and scaling types, and introduces new solutions using symmetry methods.

Contribution

It provides a complete symmetry classification and derives new similarity solutions for a specific class of fifth-order PDEs describing kink phenomena.

Findings

Lie symmetry classification of the PDE family

Derivation of travel-wave and scaling solutions

Recovery and discovery of new solutions using symmetries

Abstract

We apply the theory of infinitesimal transformations for the study of a family of 1+1 fifth-order partial differential equations which have been proposed before for the description of multiple kink solutions. In this analysis we perform a complete classification of the Lie symmetries and of the one-dimensional optimal system. The results are applied for the derivation of similarity solutions and in particular we find travel-wave and scaling solutions. We show that the kink-solution of these equations can be recovered by the use of the Lie symmetries, while new solutions are also derived.