On stability and instability of standing waves for 2d-nonlinear Schr\"odinger equations with point interaction

TL;DR

This paper investigates the existence and stability of ground-state standing waves in a 2D nonlinear Schrödinger equation with point interaction, revealing stability near the negative eigenvalue and stability or instability depending on the nonlinearity's criticality.

Contribution

It provides new insights into the stability regimes of standing waves for 2D NLS with point interaction, especially near the negative eigenvalue and for large frequencies.

Findings

Stable near the negative eigenvalue

Stable in $L^2$-subcritical/critical cases at large frequencies

Unstable in $L^2$-supercritical case at large frequencies

Abstract

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schr\"odinger equation with a point interaction and a focusing power nonlinearity. The Schr\"odinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the $L^2$-subcritical or critical case, while the instability in the $L^2$-supercritical case.