Uniqueness for the Camassa–Holm equation with non-homogeneous boundary conditions

We establish the uniqueness of solutions of the Camassa–Holm equation on a finite interval with non-homogeneous boundary conditions in the case of bounded momentum. A similar result for the higher-order Camassa–Holm system is also given. Our proofs rely on energy-type methods, with some multipliers given as solutions of some auxiliary elliptic systems.