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Orbital Stability of Periodic Traveling Waves in the $b$-Camassa-Holm Equation

Brett Ehrman, Mathew A. Johnson

Abstract

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solutions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every $b>1$, and hence applies outside of the completely integrable cases ($b=2$ and $b=3$).

Orbital Stability of Periodic Traveling Waves in the $b$-Camassa-Holm Equation

Abstract

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solutions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every , and hence applies outside of the completely integrable cases ( and ).
Paper Structure (9 sections, 10 theorems, 128 equations, 1 figure)

This paper contains 9 sections, 10 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some Basic Properties of the b-CH Family
  3. Hamiltonian Structure and Conservation Laws
  4. Existence of Periodic Traveling Waves
  5. Orbital Stability Criteria
  6. Conclusions
  7. Appendix
  8. Analysis of $\{T,\omega_1\}_{E,c}$
  9. A Technical Result

Key Result

Lemma 1

For a fixed $b>1$, let $\mu(\cdot;a,E,c)$ be a $T$-periodic solution of e:quad. Then $\mu$ is a critical point of the action functional $\Lambda(m)$ provided that

Figures (1)

  • Figure 1: Depiction of the effective potential $V(\varphi;a,c)$ for an admissible value of $a$. Note there is a vertical asymptote at $\varphi=c$ and that, and that all the periodic solutions here exist for $\varphi<c$.

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 1
  • Remark 4
  • Lemma 2
  • Lemma 3: Sylvester's Inertial Law L03
  • proof : Proof of Theorem \ref{['T:morse']}
  • ...and 15 more