On selection of standing wave at small energy in the 1D Cubic Schr\"odinger Equation with a trapping potential

TL;DR

This paper extends the analysis of small-energy standing waves in the 1D cubic Schrödinger equation with a trapping potential, incorporating multiple discrete modes and advanced virial inequalities.

Contribution

It generalizes existing theory to cases with many discrete modes using Darboux transformations and refined profile concepts.

Findings

Extended the theory to multiple discrete modes

Demonstrated the applicability to a class of repulsive potentials

Provided insights into nonlinear dissipation mechanisms

Abstract

Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associate to a repulsive potential by a series of Darboux transformations. This a simpler model than the kink stability for wave equations, but is still a classical one and retains some of the main difficulties.