Orbital stability of smooth solitary waves for the modified Camassa-Holm equation

2024/12/26 by Xijun Deng, Deng, Xijun, Stéphane Lafortune +3 · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2412.19342

Abstract

In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background k, are unique up to translation for each permissible value of k and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in H¹(ℝ).

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• Stability of 2-soliton solutions for the modified Camassa-Holm equation with cubic nonlinearity

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