Stability of smooth solitary waves in the b -Camassa–Holm equation

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 H 3 ( 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 .