Stability of smooth multi-solitons for the Camassa–Holm equation

Consideration in this paper is the stability of exact smooth multi-solitons for the Camassa–Holm equation. By constructing a suitable Lyapunov functional, it is found that the smooth multi-solitons are non-isolated constrained minimizers satisfying a suitable variational nonlocal elliptic equation and the dynamical stability issue is reduced to study of the spectrum of explicit linearized systems. Our approach in the spectral analysis consists in an invariant for the multi-solitons and new operator identities motivated by the bi-Hamilton structure of the Camassa–Holm equation. The key ingredient in the spectral analysis is to use integrable property of the recursion operator of the Camassa–Holm equation. It is demonstrated here that orbital stability of shape of smooth single soliton implies that the shapes of all smooth multi-solitons are dynamically stable under small disturbances in a suitable Sobolev space.