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Periodic Waves for the Regularized Camassa-Holm Equation: Existence and Spectral Stability

Fabio Natali

TL;DR

This work develops a rigorous framework for the existence and stability of periodic traveling waves in the regularized Camassa-Holm equation on a periodic domain. Using Crandall–Rabinowitz bifurcation theory and a global continuation argument (à la Buffoni–Toland), the authors construct a continuous curve of zero-mean periodic waves $c o oldsymbol{phi}_c$ for speeds $c> rac{oldsymbol{omega}}{2}$, including explicit small-amplitude expansions. They then perform a spectral analysis of the linearized operator, reducing the problem to a Hill/Schrödinger operator to count negative and zero eigenvalues in the zero-mean subspace, and show that there is exactly one negative eigenvalue and a simple zero eigenvalue (spanned by $oldsymbol{phi}'$) for the restricted operator. Under a nondegeneracy condition (nonvanishing $d_c$), they prove spectral stability and, via the Grillakis–Shatah–Strauss/ANP framework, orbital stability of the waves in $H^1_{ ext{per},m}$. The results sharpen the understanding of wave stability for the rCH model and clarify the role of the drift parameter in avoiding fold points that occur in the CH limit.

Abstract

In this paper, we investigate the existence and spectral stability of periodic traveling wave solutions for the regularized Camassa-Holm equation. To establish the existence of periodic waves, we employ tools from bifurcation theory to construct solutions with the zero-mean property. We also prove that such waves may not exist for the well-known Camassa-Holm equation. Regarding spectral stability, we analyze the difference between the number of negative eigenvalues of the second variation of the Lyapunov functional at the wave, restricted to the space of zero-mean periodic functions, and the number of negative eigenvalues of the matrix formed from the tangent space associated with the low-order conserved quantities of the evolution model. Finally, we address the problem of orbital stability as a consequence of the spectral stability.

Periodic Waves for the Regularized Camassa-Holm Equation: Existence and Spectral Stability

TL;DR

This work develops a rigorous framework for the existence and stability of periodic traveling waves in the regularized Camassa-Holm equation on a periodic domain. Using Crandall–Rabinowitz bifurcation theory and a global continuation argument (à la Buffoni–Toland), the authors construct a continuous curve of zero-mean periodic waves for speeds , including explicit small-amplitude expansions. They then perform a spectral analysis of the linearized operator, reducing the problem to a Hill/Schrödinger operator to count negative and zero eigenvalues in the zero-mean subspace, and show that there is exactly one negative eigenvalue and a simple zero eigenvalue (spanned by ) for the restricted operator. Under a nondegeneracy condition (nonvanishing ), they prove spectral stability and, via the Grillakis–Shatah–Strauss/ANP framework, orbital stability of the waves in . The results sharpen the understanding of wave stability for the rCH model and clarify the role of the drift parameter in avoiding fold points that occur in the CH limit.

Abstract

In this paper, we investigate the existence and spectral stability of periodic traveling wave solutions for the regularized Camassa-Holm equation. To establish the existence of periodic waves, we employ tools from bifurcation theory to construct solutions with the zero-mean property. We also prove that such waves may not exist for the well-known Camassa-Holm equation. Regarding spectral stability, we analyze the difference between the number of negative eigenvalues of the second variation of the Lyapunov functional at the wave, restricted to the space of zero-mean periodic functions, and the number of negative eigenvalues of the matrix formed from the tangent space associated with the low-order conserved quantities of the evolution model. Finally, we address the problem of orbital stability as a consequence of the spectral stability.
Paper Structure (7 sections, 13 theorems, 87 equations, 2 figures)

This paper contains 7 sections, 13 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existence of periodic traveling waves - Proof of Theorem $\ref{['theorem-stability']}-(i)$
  3. The linearized operator - Proof of Theorem $\ref{['theorem-stability']}-(ii)$
  4. The non-positive spectrum of $\mathcal{L}_{\Pi}$.
  5. The behaviour of the function $A(c)$.
  6. Spectral stability of periodic waves - Proof of Theorem $\ref{['theorem-stability']}-(iii)$
  7. A remark on the orbital stability of periodic waves

Key Result

Theorem 1.5

Let $c>\frac{\omega}{2}$ be fixed. (i) There exists a continuous mapping $c\in \left(\frac{\omega}{2},+\infty\right) \mapsto \phi_c=\phi\in H_{\rm per,m}^{\infty}$ of $2\pi-$periodic functions that solves equation $(CHode)$ with constant $A$ given by $(constA)$. (ii) The linear operator $\mathcal{L}

Figures (2)

  • Figure 3.1: Graph of $A$ for $\omega=1$ (left), and graph of $A$ for $\omega=2$ (right).
  • Figure 3.2: Graph of $A$ for $\omega=3$ (left), and graph of $A$ for $\omega=5$ (right).

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 24 more