On asymptotic stability in energy space of ground states for Nonlinear Schr\"odinger equations

TL;DR

This paper proves that solutions near stable ground states of nonlinear Schrödinger equations in higher dimensions asymptotically decompose into a ground state plus dispersive waves, under the Fermi Golden Rule condition.

Contribution

It extends previous results by relaxing the sign condition in the Fermi Golden Rule hypothesis for asymptotic stability analysis.

Findings

Solutions converge to a ground state plus dispersive wave

Improved the sign condition in the FGR hypothesis

Applicable to nonlinear Schrödinger equations in 3 or higher dimensions

Abstract

We consider nonlinear Schr\"odinger equations in dimension 3 or higher. We prove that symmetric finite energy solutions close to orbitally stable ground states converge asymptotically to a sum of a ground state and a dispersive wave assuming the so called Fermi Golden Rule (FGR) hypothesis. We improve the sign condition required in a recent paper by Gang Zhou and I.M.Sigal