On the inelastic 2-soliton collision for gKdV equations with general nonlinearity

TL;DR

This paper investigates 2-soliton collisions in generalized KdV equations, identifying which nonlinearities lead to elastic or inelastic collisions, and confirms that only integrable cases exhibit perfectly elastic interactions.

Contribution

It classifies nonlinearities in gKdV equations that result in elastic versus inelastic 2-soliton collisions, extending previous research and confirming integrability as the key factor.

Findings

Only quadratic, cubic, and Gardner nonlinearities produce elastic collisions.

Non-integrable nonlinearities lead to inelastic 2-soliton interactions.

The study completes the classification of soliton collision behaviors in gKdV equations.

Abstract

We study the problem of 2-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle. We classify the nonlinearities for which collisions are elastic or inelastic. Our main result states that in the case of small solitons, with one soliton smaller than the other one, the unique nonlinearities allowing a perfectly elastic collision are precisely the integrable cases, namely the quadratic (KdV), cubic (mKdV) and Gardner nonlinearities.