Symmetries and integrability of the 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 $$\begin{aligned} u_{t}+k(u-u_{xx})^2u_{x}-u_{xxt}+(u^{2}-{u_{x}}^2)(u_{x}-u_{xxx})=0. \end{aligned}$$The commutator table and adjoint representation of the symmetries are presented to construct one-dimensional optimal system. By using the one-dimensional optimal system, we reduce the order or number of independent variables of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally, we apply the Painlevé test to the resultant nonlinear ordinary differential equation and it is observed that the equation is integrable.