Orbital stability of solitary waves and a Liouville-type property to the cubic Camassa–Holm-type equation

Abstract

We classify the existence and non-existence of a class of localized solitary waves for the cubic Camassa–Holm-type equation which arises as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves. In addition to those peaked solitary-wave solutions, we show by the phase portrait method that there are smooth and cusped solitary waves according to the wave speed and coefficients of nonlocal linear and nonlinear dispersions. We then prove that the smooth solitary-wave solutions are orbitally stable in the energy space under small perturbation. Finally, we establish a Liouville-type property of the solution.