Inelastic interaction of nearly equal solitons for the BBM equation

TL;DR

This paper investigates the inelastic interaction of nearly equal solitons for the BBM equation, demonstrating soliton preservation, separation, and inelastic collisions except in the integrable case, extending previous results from gKdV equations.

Contribution

It extends prior work on soliton interactions to the BBM equation, showing soliton preservation, separation, and inelastic collision behavior in a non-integrable setting.

Findings

Solitons are preserved during interaction.

Solitons remain separated by large distances over time.

Collision is inelastic unless the equation is integrable.

Abstract

This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of the results obtained in arXiv:0910.3204 by the same authors, addressing the same question for the quartic (gKdV) equation. First, we prove that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable (KdV) equation in this regime. Second, we prove that the collision is not perfectly elastic, except in the integrable case (i.e. in the limiting case of the (KdV) equation).