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

Xijun Deng, Stéphane Lafortune, Zhisu Liu

TL;DR

The paper establishes orbital stability for smooth solitary waves of the modified Camassa–Holm equation on a nonzero background by recasting the equation in the momentum-variable Hamiltonian framework with three conserved functionals F1,F2,F3. Solitary waves are shown to exist for given speed c and background k, are variationally characterized as critical points of an action Λ, and possess a Hessian with a single negative eigenvalue and a translation zero mode. The Vakhitov–Kolokolov condition is derived and shown to hold analytically, guaranteeing positive definiteness of the constrained Hessian and hence stability in H^1 for m-perturbations and, equivalently, in H^3 for the corresponding u formulation. The results extend the stability theory for mCH by handling cubic nonlinearity on nonzero backgrounds, with F2 and F3 explicitly identified as Casimirs and incorporated into a Lyapunov functional that enforces the stability result.

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 $H^1(\mathbb{R})$.

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

TL;DR

The paper establishes orbital stability for smooth solitary waves of the modified Camassa–Holm equation on a nonzero background by recasting the equation in the momentum-variable Hamiltonian framework with three conserved functionals F1,F2,F3. Solitary waves are shown to exist for given speed c and background k, are variationally characterized as critical points of an action Λ, and possess a Hessian with a single negative eigenvalue and a translation zero mode. The Vakhitov–Kolokolov condition is derived and shown to hold analytically, guaranteeing positive definiteness of the constrained Hessian and hence stability in H^1 for m-perturbations and, equivalently, in H^3 for the corresponding u formulation. The results extend the stability theory for mCH by handling cubic nonlinearity on nonzero backgrounds, with F2 and F3 explicitly identified as Casimirs and incorporated into a Lyapunov functional that enforces the stability result.

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 , are unique up to translation for each permissible value of 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 .
Paper Structure (11 sections, 10 theorems, 99 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Smooth solitary wave solutions
  4. Existence of smooth solitary wave solutions
  5. Variational characterization
  6. Spectral properties of $\mathcal{L}$
  7. Orbital stability analysis
  8. Analysis of the Vakhitov-Kolokolov condition
  9. Verification of the stability criterion \ref{['4.9']}
  10. Proof of Theorem 1.1
  11. Conservation of the functional $F_1$ and $F_2$

Key Result

Theorem 1.1

For fixed $c>0$, and $k\in(\frac{\sqrt{c}}{3}, \frac{\sqrt{3c}}{3})$, there exists a unique smooth solitary wave $m(t, x) =\mu (x - ct)$ of the mCH equation 1. This solitary wave $\mu (x - ct)$ is orbitally stable in the space $X_k$ defined in 4, namely, if for every $\varepsilon>0$ there exists $\d

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • ...and 17 more