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

TL;DR

This paper proves the orbital stability of a train of N solitary waves in the two-component Camassa-Holm system, demonstrating their robustness in the energy space through modulation theory and energy estimates.

Contribution

It introduces a novel stability proof for multiple solitary waves in the two-component Camassa-Holm system using advanced modulation and energy methods.

Findings

Train of N solitary waves is orbitally stable in H^1 x L^2 space.

Solitary waves interact like solitons and are sufficiently decoupled.

Method combines almost monotonicity of local energy with stability analysis.

Abstract

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 $H^1(\R)\times L^2(\R)$.