Stability of smooth multisolitons for the two-component Camassa–Holm system

In this work, we study the stability of exact smooth multisolitons for the two-component Camassa–Holm system, a completely integrable model for unidirectional shallow-water waves. By analyzing the spectrum of the linearized system and using a suitable Lyapunov functional, we show that these smooth multisolitons, as nonisolated constrained minimizers, are dynamically stable under small perturbations in an appropriate Sobolev space. A key part of our analysis involves employing the recursion operator from the bi-Hamiltonian structure and introducing a new transformation to diagonalize the nondiagonal matrix form in the second variation of the Lyapunov functional. This approach is essential due to the significant differences compared to the classical Camassa–Holm equation.