Ground states of the two-dimensional dispersion managed nonlinear Schr\"odinger equation

TL;DR

This paper investigates the existence of ground states in a two-dimensional dispersion managed nonlinear Schrödinger equation, revealing a mass-dependent threshold for minimizer existence and identifying extremizers of related inequalities.

Contribution

It establishes a threshold phenomenon for minimizer existence based on mass and characterizes extremizers of associated inequalities.

Findings

Existence of minimizers depends on a critical mass threshold.

Identification of extremizers for the Gagliardo-Nirenberg-Strichartz inequality.

Analysis applies to all power-law nonlinearities, including at the threshold.

Abstract

We consider the variational problem with a mass constraint arising from the two-dimensional dispersion managed nonlinear Schr\"odinger equation with power-law type nonlinearity. We prove a threshold phenomenon with respect to mass for the existence of minimizers for all possible powers of nonlinearities, including at the threshold itself. This threshold is closely related to the best constant for the Gagliardo-Nirenberg-Strichartz type inequality whose extremizers are found as a byproduct.