On smoothness and uniqueness of multi-solitons of the non-linear Schr{\"o}dinger equations

TL;DR

This paper investigates the smoothness and uniqueness of multi-solitons in non-linear Schrödinger equations, establishing conditions under which these solutions are smooth and unique, depending on the non-linearity and stability properties.

Contribution

It demonstrates that multi-solitons are smooth based on the non-linearity's regularity and proves uniqueness in specific stability or mass-critical cases.

Findings

Multi-solitons are smooth if the non-linearity is regular.

Uniqueness holds when ground states are stable or in the mass-critical case.

Results extend understanding of solution properties in non-linear Schrödinger equations.

Abstract

In this paper, we study some properties of multi-solitons for the non-linear Schr{\"o}dinger equations in R^d with general non-linearities. Multi-solitons have already been constructed in H^1, successively by Merle, by Martel and Merle, and by C{\^o}te, Martel and Merle. We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.