Collision of solitons for a non-homogenous version of the KdV equation

TL;DR

This paper studies soliton interactions in a non-homogeneous KdV-type equation with small dispersion, revealing that solitons behave like classical KdV solitons at leading order and identifying an instability scenario affecting soliton stability and collisions.

Contribution

It introduces a non-homogeneous KdV model with small dispersion and analyzes soliton interactions, including a novel instability scenario affecting soliton stability and collisions.

Findings

Solitons interact like classical KdV solitons at leading order.

A scenario where short-wave solitons remain stable while long-wave parts become unstable.

Perturbations can prevent soliton collisions.

Abstract

We consider KdV-type equations with $C^1$ nonhomogeneous nonlinearities and small dispersion $\varepsilon$. The first result consists in the conclusion that, in the leading term with respect to $\varepsilon$, the solitary waves in this model interact like KdV solitons. Next it turned out that there exists a very interesting scenario of instability in which the short-wave soliton remains stable whereas a small long-wave part, generated by perturbations of original equation, turns to be unstable, growing and destroying the leading term. At the same time, such perturbation can eliminate the collision of solitons.