Symmetry of traveling wave solutions for a Camassa–Holm type equation with higher-order nonlinearity

Abstract We are concerned with a Camassa–Holm type equation with higher-order nonlinearity including some integrable peakon models such as the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We show that all the horizontal symmetric waves for this equation must be traveling waves. This extends the previous results for the Camassa–Holm and Novikov equations.