Stability of the train of N solitary waves for the two-component Camassa–Holm shallow water system

Considered herein is the integrable two-component Camassa–Holm shallow water system derived in the context of shallow water theory, which admits blow-up solutions and the solitary waves interacting like solitons. Using modulation theory, and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave, we prove that the train of N   solitary waves, which are sufficiently decoupled, is orbitally stable in the energy space H1(R)×L2(R)H1(R)×L2(R).