Stability of smooth solitary waves in the [formula omitted]-Camassa–Holm equation

Abstract

We derive the precise stability criterion for smooth solitary waves in the b-family of Camassa–Holm equations. The smooth solitary waves exist on the constant background. In the integrable cases b=2 and b=3, we show analytically that the stability criterion is satisfied and smooth solitary waves are orbitally stable with respect to perturbations in H3(R). In the non-integrable cases, we show numerically and asymptotically that the stability criterion is satisfied for every b>1. The orbital stability theory relies on a Hamiltonian formulation of the b-family which is different from the Hamiltonian formulations available for b=2 and b=3.