Orbital stability of smooth solitons in H1 ∩ W1,4 for the modified Camassa-Holm equation

Abstract

We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the $H^1\cap W^{1,4}$ stability result in Li and Liu (2021) [8].