Asymptotic stability of ground states in 2D nonlinear Schr\"odinger equation including subcritical cases

TL;DR

This paper proves the asymptotic stability of ground states in 2D nonlinear Schrödinger equations with a broad class of nonlinearities, including subcritical and supercritical cases, for small initial data.

Contribution

It introduces a general method to establish stability of bound states in 2D NLS with diverse nonlinearities, extending previous results to broader cases.

Findings

Solutions with small initial data converge to a nonlinear bound state

Nonlinear bound states are proven to be asymptotically stable

Dispersive estimates are developed for the linearized dynamics around bound states

Abstract

We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that "shadows" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.