Orbital stability of peakons for a generalization of the modified Camassa–Holm equation

The orbital stability of the peaked solitary-wave solutions for a generalization of the modified Camassa–Holm equation with both cubic and quadratic nonlinearities is investigated. The equation is a model of asymptotic shallow-water wave approximations to the incompressible Euler equations. It is also formally integrable in the sense of the existence of a Lax formulation and bi-Hamiltonian structure. It is demonstrated that, when the Camassa–Holm energy counteracts the effect of the modified Camassa–Holm energy, the peakon and periodic peakon solutions are orbitally stable under small perturbations in the energy space.