Asymptotic integrability and Hamilton theory of soliton's motion along large-scale background waves

TL;DR

This paper develops a Hamiltonian framework for understanding soliton interactions with large-scale background waves in asymptotically integrable systems, applying it specifically to the Kaup-Boussinesq system.

Contribution

It introduces a universal Hamiltonian formulation for soliton dynamics along background waves in asymptotically integrable equations, linking it to Riemann invariants.

Findings

Derived Hamilton equations for soliton motion in background waves.

Applied the theory to the Kaup-Boussinesq system.

Provided explicit expressions for physical properties via Riemann invariants.

Abstract

We consider the problem of soliton-mean field interaction for the class of asymptotically integrable equations, where the notion of the asymptotic integrability means that the Hamilton equations for the high-frequency wave packet's propagation along a large-scale background wave have an integral of motion. Using the Stokes remark, we transform this integral to the integral for the soliton's equations of motion and then derive the Hamilton equations for the soliton's dynamics in a universal form expressed in terms of the Riemann invariants for the hydrodynamic background wave. The physical properties are specified by the concrete expressions for the Riemann invariants. The theory is illustrated by its application to the soliton's dynamics which is described by the Kaup-Boussinesq system.