Multi-existence of multi-solitons for the supercritical nonlinear Schr\"odinger equation in one dimension

TL;DR

This paper constructs an N-parameter family of N-solitons for the supercritical nonlinear Schrödinger equation in one dimension, extending previous work on similar solutions for related equations.

Contribution

It demonstrates the existence of multi-soliton solutions for the supercritical nonlinear Schrödinger equation in one dimension, without establishing uniqueness or classification.

Findings

Constructed N-parameter family of N-solitons

Extended previous results from Korteweg-de Vries to Schrödinger equation

No classification or uniqueness results obtained

Abstract

For the L2 supercritical generalized Korteweg-de Vries equation, we proved in a previous article the existence and uniqueness of an N-parameter family of N-solitons. Recall that, for any N given solitons, we call N-soliton a solution of the equation which behaves as the sum of these N solitons asymptotically as time goes to infinity. In the present paper, we also construct an N-parameter family of N-solitons for the supercritical nonlinear Schr\"odinger equation, in dimension 1 for the sake of simplicity. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.