Improved uniqueness of multi-breathers of the modified Korteweg-de Vries equation

TL;DR

This paper improves the understanding of the uniqueness of multi-breather solutions in the modified Korteweg-de Vries equation, extending previous results to cases with at most one non-positive velocity.

Contribution

It extends the uniqueness result for multi-breathers of mKdV to include cases with at most one zero or negative velocity, broadening the class of solutions with guaranteed uniqueness.

Findings

Uniqueness holds if at most one breather has non-positive velocity.

Previous results required all velocities to be positive or faster convergence.

The result applies to a broader set of multi-breather configurations.

Abstract

We consider multi-breathers of (mKdV). Previously, a smooth multi-breather was constructed, and proved to be unique in two cases: first, if the class of super-polynomial convergence to the profile, and second, under the assumption that all speeds of the breathers involved are positive (without rate of convergence). The goal of this short note is to improve the second result: we show that uniqueness still holds if at most one velocity is negative or zero.