On an intercritical log-modified nonlinear Schrodinger equation in two spatial dimensions

TL;DR

This paper studies a two-dimensional nonlinear Schrödinger equation with a logarithmic correction, modeling quantum droplets, proving global solutions, ground state existence, and stability properties.

Contribution

It introduces and analyzes a novel intercritical log-modified nonlinear Schrödinger equation, establishing global well-posedness, ground state existence, and stability results.

Findings

Global existence of strong solutions in energy space

Existence and uniqueness of nonlinear ground states

Orbital stability of energy minimizers

Abstract

We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available.