Orbital stability of peakon solutions for a generalized higher-order Camassa–Holm equation

In this paper, we investigate the orbital stability issue of a generalized higher-order Camassa–Holm (HOCH) equation, which is a higher-order extension of the quadratic CH equation. Firstly, we show that the HOCH equation admits a global weak peakon solution by paring it with some smooth test function. Secondly, with the help of two conserved quantities and the non-sgn-changing condition, we prove the orbital stability of this peakon solution in the energy space in the sense that its shape remains approximately the same for all times. Our results enrich the research of the orbital stability for the CH-type equations and are useful to better understand the impact of higher-order nonlinearities on the dispersion dynamics.