On instability of some approximate periodic solutions for the full nonlinear Schr\"odinger equation

TL;DR

This paper proves the asymptotic stability of certain nonlinear bound states in the quintic nonlinear Schrödinger equation, showing that some approximate periodic solutions do not persist in the full nonlinear setting.

Contribution

It introduces a Fermi Golden Rule analysis to establish stability of asymmetric bound states, advancing understanding of solution persistence in nonlinear Schrödinger equations.

Findings

Asymptotic stability of asymmetric bound states proved

Approximate periodic solutions do not persist in the full NLS

Fermi Golden Rule analysis applied to nonlinear Schrödinger operator

Abstract

Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schr\"odinger operator with symmetric potential. This goes in the direction of proving that the approximate periodic solutions of the NLS shadowed in a recent work of the second author with Michael Weinstein do not persist for the full NLS.