Stability of peakons for the generalized modified Camassa-Holm equation

Abstract

In this paper, we consider the orbital stability of peakons for the generalized modified Camassa-Holm (CH) equation, which admits a single peaked soliton and multi-peakon solutions. We firstly apply the definition of weak solution and some combination formulas to investigate the existence of single peakon. Then we prove two useful conservation laws which play a key role in our proof of stability of peakons. Finally by introducing a new auxiliary functional $h$ and establishing a crucial polynomial inequality, we successfully extend the result of stability of peakons for the modified CH equation (Comm. Math. Phys., 322:967-997, 2013) to the generalized case when $n$ is any positive odd integer.