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...

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Bibliographic Details
Published in:Physica. D 2022-11, Vol.440, p.133477, Article 133477
Main Authors: Lafortune, Stéphane, Pelinovsky, Dmitry E.
Format: Article
Language:English
Subjects:
Camassa–Holm equation
Integrability
Stability of solitary waves
Traveling wave solutions
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Summary: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. •We derive a stability criterion for smooth solitary waves of the b-family.•In the integrable cases, we show that the stability criterion is satisfied.•We show numerically and asymptotically that the criterion is satisfied if b>1.•We use a Hamiltonian formulation of the b-family.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2022.133477