Advances in Differential Equations

On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations

Liangchen Li, Hengyan Li, and Weiping Yan

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Abstract

This paper mainly studies the explicit wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the Camassa-Holm and Degasperis-Procesi equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime. We prove that the Camassa-Holm and Degasperis-Procesi equations admit stable explicit self-similar solutions. After that, the nonlinear instability of explicit self-similar solution for the Korteweg-de Vries equation is given.

Article information

Source
Adv. Differential Equations, Volume 25, Number 5/6 (2020), 315-334.

Dates
First available in Project Euclid: 16 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.ade/1589594421

Mathematical Reviews number (MathSciNet)
MR4099222

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics 35A21: Propagation of singularities 35B35: Stability

Citation

Li, Liangchen; Li, Hengyan; Yan, Weiping. On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations. Adv. Differential Equations 25 (2020), no. 5/6, 315--334. https://projecteuclid.org/euclid.ade/1589594421


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