Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion

TL;DR

This paper models interactions between bore-like structures and solitons in shallow water using a coupled Burgers-KdV system, revealing complex behaviors like refraction, reflection, and fusion through analytical and numerical methods.

Contribution

It introduces a novel coupled PDE system derived via symmetry and variational principles, providing exact solutions and analyzing complex nonlinear interactions.

Findings

Exact compound soliton solutions obtained.

Numerical simulations show refraction, reflection, and fusion behaviors.

System tends toward an integrable Gardner equation at equilibrium.

Abstract

We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system are presented, based on Lie group symmetry and reduced variational principles. Exact compound soliton solutions are obtained, and numerical simulations show that the Burgers and KdV momenta tend toward a balance at which the coupled system reduces to the integrable Gardner equation. The numerical simulations also reveal rich nonlinear solution behaviours that include refraction, reflection, and soliton fusion, before the balance is finally achieved.