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

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.