Propagation of generalized Korteweg-de Vries solitons along large-scale waves

TL;DR

This paper studies how solitons propagate along large-scale background waves in the generalized Korteweg-de Vries equation, deriving Hamiltonian equations for their motion and validating results with numerical solutions.

Contribution

It introduces a Hamiltonian framework for soliton motion along background waves in gKdV and provides simple velocity relationships based on local wave values.

Findings

Soliton paths match numerical solutions closely.

Derived explicit relationships for soliton velocity.

Background wave evolution remains unaffected by soliton motion.

Abstract

We consider propagation of solitons along large scale background waves in the generalized Korteweg-de Vries (gKdV) equation theory when the width of the soliton is mach smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equations for soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.