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Asymptotic stability of smooth solitons and multi-solitons for the Camassa--Holm equation

Robin Ming Chen, Yang Lan, Yue Liu, Zhong Wang

TL;DR

The paper addresses the problem of proving asymptotic stability for smooth Camassa–Holm solitons and multi-solitons in the energy space $H^1(\mathbb{R})$. It develops a novel, integrability-driven spectral framework based on the recursion operator and squared eigenfunctions to obtain a complete linear spectral resolution and exponential decay in weighted spaces, which substitutes for the lack of a classical Schrödinger framework. A nonlinear Liouville (rigidity) property then converts linear decay into nonlinear asymptotic stability, extending from single solitons to well-separated trains and tying into the explicit multi-soliton solutions from inverse scattering. The results solidify the soliton resolution picture for CH by proving asymptotic convergence to a modulated soliton or soliton train, and they offer a unified perspective that also informs the linear theories for KdV and mKdV via squared-eigenfunction expansions. Overall, the work provides a rigorous, integrability-based pathway to asymptotic stability in a nonlocal, nontrivial dispersive system with broad implications for soliton dynamics in integrable PDEs.

Abstract

We establish the asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm (CH) equation in the energy space $H^1(\R)$. We show that solutions initially close to a soliton converge, up to translation, weakly in $H^1(\R)$ as time tends to infinity to a (possibly different) soliton. The analysis is based on a Liouville-type rigidity theorem characterizing solutions that remain localized near a soliton trajectory. A central feature of the proof is a complete spectral resolution of the linearized CH operator around a soliton. This linear theory is obtained via the bi-Hamiltonian and integrable structure of the CH equation, through the recursion operator and the completeness of the associated squared eigenfunctions. It provides a substitute for the classical spectral framework used in KdV and gKdV equations, which is unavailable in the nonlocal and variable-coefficient setting of CH. The spectral resolution yields sharp decay estimates for the linearized flow in exponentially weighted spaces, which in turn lead to the nonlinear rigidity result and the asymptotic stability of a single soliton. Combined with known orbital stability results, this approach extends to well-ordered trains of solitons and to the explicit multi-soliton solutions generated by the inverse scattering method. As an additional application, we revisit the linearized problems associated with other integrable dispersive equations, including the KdV and mKdV equations, from the perspective of squared-eigenfunction expansions.

Asymptotic stability of smooth solitons and multi-solitons for the Camassa--Holm equation

TL;DR

The paper addresses the problem of proving asymptotic stability for smooth Camassa–Holm solitons and multi-solitons in the energy space . It develops a novel, integrability-driven spectral framework based on the recursion operator and squared eigenfunctions to obtain a complete linear spectral resolution and exponential decay in weighted spaces, which substitutes for the lack of a classical Schrödinger framework. A nonlinear Liouville (rigidity) property then converts linear decay into nonlinear asymptotic stability, extending from single solitons to well-separated trains and tying into the explicit multi-soliton solutions from inverse scattering. The results solidify the soliton resolution picture for CH by proving asymptotic convergence to a modulated soliton or soliton train, and they offer a unified perspective that also informs the linear theories for KdV and mKdV via squared-eigenfunction expansions. Overall, the work provides a rigorous, integrability-based pathway to asymptotic stability in a nonlocal, nontrivial dispersive system with broad implications for soliton dynamics in integrable PDEs.

Abstract

We establish the asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm (CH) equation in the energy space . We show that solutions initially close to a soliton converge, up to translation, weakly in as time tends to infinity to a (possibly different) soliton. The analysis is based on a Liouville-type rigidity theorem characterizing solutions that remain localized near a soliton trajectory. A central feature of the proof is a complete spectral resolution of the linearized CH operator around a soliton. This linear theory is obtained via the bi-Hamiltonian and integrable structure of the CH equation, through the recursion operator and the completeness of the associated squared eigenfunctions. It provides a substitute for the classical spectral framework used in KdV and gKdV equations, which is unavailable in the nonlocal and variable-coefficient setting of CH. The spectral resolution yields sharp decay estimates for the linearized flow in exponentially weighted spaces, which in turn lead to the nonlinear rigidity result and the asymptotic stability of a single soliton. Combined with known orbital stability results, this approach extends to well-ordered trains of solitons and to the explicit multi-soliton solutions generated by the inverse scattering method. As an additional application, we revisit the linearized problems associated with other integrable dispersive equations, including the KdV and mKdV equations, from the perspective of squared-eigenfunction expansions.
Paper Structure (27 sections, 38 theorems, 360 equations)

This paper contains 27 sections, 38 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. CH solitons
  3. Nonlinear stability of smooth CH solitons
  4. Main results
  5. Challenges and methodology
  6. Organization of the paper
  7. Preliminaries
  8. Eigenvalue problem, conservation laws and multi-solitons
  9. Recursion operators and completeness relations of the squared eigenfunctions
  10. The recursion operators at $N$-solitons
  11. Well-posedness theory
  12. Monotonicity formula and exponential decay of $H^1$-localized solutions
  13. Modulation theory
  14. Nonlinear Liouville property implies asymptotic stability
  15. Construction of the limit solution.
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $c>2\omega$, $u_0\in Y_+$ and $u\in \mathcal{C}(\mathbb{R};H^1(\mathbb{R}))$ be the solution of eq1 emanating from $u_0$. There exists $\alpha_0>0$ such that if $\|u_0-\varphi_c\|_{H^1}<\alpha_0$ and $u(t)$ is $H^1$-localized, then there exist $y_0\in\mathbb{R}$ and $c^\star>2\omega$ such that

Theorems & Definitions (73)

  • Definition 1: EM07
  • Theorem 1.1: Rigidity property for the CH flow
  • Theorem 1.2: Asymptotic stability of smooth solitons
  • Theorem 1.3: Asymptotic stability of well-ordered soliton trains
  • Corollary 1.4: Asymptotic stability of smooth multi-solitons
  • Lemma 2.1: WL20
  • proof
  • Remark 2.1
  • Lemma 2.2: Spectrum of $\mathcal{R}(\varphi)$
  • proof
  • ...and 63 more