Symmetries and Integrability of Modified Camassa-Holm Equation with an Arbitrary Parameter

TL;DR

This paper investigates the symmetry properties and integrability of a modified Camassa-Holm equation with an arbitrary parameter, deriving solutions and testing for integrability using Lie symmetries and Painlevé analysis.

Contribution

It introduces a novel analysis of the modified Camassa-Holm equation with an arbitrary parameter, including symmetry reduction, solution derivation, and integrability testing.

Findings

Reduced the equation's order using Lie symmetries

Obtained novel solutions for the reduced ODEs

Applied Painlevé test to assess integrability

Abstract

We study the symmetry and integrability of a modified Camassa-Holm Equation (MCH), with an arbitrary parameter $k,$ of the form $$u_{t}+k(u-u_{xx})^2u_{x}-u_{xxt}+(u^{2}-{u_{x}}^2)(u_{x}-u_{xxx})=0.$$ By using Lie point symmetries we reduce the order of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally we apply the Painlev\'e Test to the resultant nonlinear ordinary differential equation.