Non dispersive solutions of the generalized KdV equations are typically multi-solitons

TL;DR

This paper proves that non-dispersive solutions of the generalized KdV equations that stay close to multi-solitons are actually multi-solitons themselves, providing a characterization for KdV and mKdV equations.

Contribution

It establishes that non-dispersive solutions near multi-solitons must be pure multi-solitons, offering a new characterization based on non-dispersion properties.

Findings

Non-dispersive solutions are necessarily multi-solitons.

Characterization of multi-solitons and multi-breathers via non-dispersion.

Results apply specifically to KdV and mKdV equations.

Abstract

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non-dispersion.