Stability of smooth solitary waves in the b-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 $H^3(\mathbb{R})$. 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.

arXiv comment19 pages; 2 figures

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Cited by in corpus (1)

• Orbital stability of smooth solitary waves for the $b$-family of Camassa-Holm equations