High speed excited multi-solitons in nonlinear Schr\"odinger equations

TL;DR

This paper constructs multi-soliton solutions in higher-dimensional nonlinear Schrödinger equations using excited states, revealing their stability properties and non-uniqueness in unstable cases.

Contribution

It introduces a method to build multi-solitons from excited states in higher dimensions and analyzes their stability and uniqueness.

Findings

Multi-solitons can be constructed from excited states in higher dimensions.

Unstable excited states lead to non-unique and unstable multi-solitons.

The solutions spread rapidly as time increases.

Abstract

We consider the nonlinear Schr\"odinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable.