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 $H^3(\mathbb{R})$. 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 different Hamiltonian formulation compared to the Hamiltonian formulations available in the integrable cases