Lie symmetries and singularity analysis for generalized shallow-water equations

TL;DR

This paper uses Lie symmetries and singularity analysis to study generalized shallow-water equations, revealing their integrability and symmetry properties, and classifying their similarity solutions.

Contribution

It provides a complete symmetry classification and similarity solution analysis for generalized Camassa-Holm and Benjamin-Bono-Mahoney equations, demonstrating their integrability.

Findings

The equations are invariant under the same three-dimensional Lie algebra.

The equations pass the singularity test indicating integrability.

Complete classification of similarity solutions was achieved.

Abstract

We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm equation and the generalized Benjamin-Bono-Mahoney equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the Camassa-Holm equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.