On the stability of ground states in 4D and 5D nonlinear Schrodinger equation including subcritical cases

TL;DR

This paper studies the stability of ground states in 4D and 5D nonlinear Schrödinger equations with various nonlinearities, showing that small initial data solutions tend to bound states using dispersive estimates.

Contribution

It extends stability analysis of ground states to higher dimensions and more general nonlinearities, including subcritical and supercritical cases, using new dispersive estimates.

Findings

Center manifold of bound states is an attractor for small initial data.

Dispersive estimates are established for linearized dynamics in higher dimensions.

Results extend previous work to rougher nonlinearities and different linear dispersive behaviors.

Abstract

We consider a class of nonlinear Schrodinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a small initial data. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that "shadows" the nonlinear evolution of the system. The methods we employ are an extension to higher dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our previous results [10, 11, 7].