Fast-moving finite and infinite trains of solitons for nonlinear Schr\"odinger equations

TL;DR

This paper proves the existence and uniqueness of both finite and infinite trains of solitons for nonlinear Schrödinger equations, especially when solitons move at large relative speeds, including complex configurations like kinks.

Contribution

It introduces new methods to construct and analyze infinite and multi-soliton trains with large relative speeds in NLS, extending previous finite soliton results.

Findings

Existence of infinite soliton trains under large relative speeds

Uniqueness of these trains in certain neighborhoods

Construction of multi-solitons with non-zero backgrounds

Abstract

We study *infinite soliton trains* solutions of nonlinear Schr\"odinger equations (NLS), i.e. solutions behaving at large time as the sum of infinitely many solitary waves. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighborhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).