Finite and infinite soliton and kink-soliton trains of nonlinear Schr\"odinger equations

TL;DR

This paper reviews multi-solitons in dispersive PDEs and presents new results on the existence and uniqueness of infinite soliton trains, including half-kinks in 1D nonlinear Schrödinger equations, under large relative speeds.

Contribution

It introduces the first rigorous proof of infinite soliton trains with large relative speeds and extends to include half-kinks in one-dimensional cases.

Findings

Proved existence and uniqueness of multi-soliton trains with infinitely many solitons.

Extended results to include half-kinks in 1D nonlinear Schrödinger equations.

Demonstrated behavior of multi-solitons as sums of weakly-interacting solitary waves.

Abstract

We will first review known results on multi-solitons of dispersive partial differential equations, which are special solutions behaving like the sum of many weakly-interacting solitary waves. We will then describe our recent joint work with Dong Li on nonlinear Schr\"odinger equations: Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of a soliton train which is a multi-soliton composed of infinitely many solitons. In the 1D case, we can add to the infinite train an additional half-kink, which is a solution with a non-zero background at minus infinity.