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

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.