Instability of bound states of a nonlinear Schr\"odinger equation with a Dirac potential

TL;DR

This paper analyzes the stability of standing waves in a nonlinear Schrödinger equation with a point defect, revealing conditions for stability, instability, and the nature of these instabilities through analytical and numerical methods.

Contribution

It introduces a perturbation and continuation approach to compute eigenvalues and characterizes stability regimes for various defect and nonlinearity types.

Findings

Stable in $ uh$ and unstable in $$ for repulsive defect under subcritical nonlinearity

Proves blowup instability under critical or supercritical nonlinear interactions

Unstable bound states in non-radial repulsive cases drift away from the defect

Abstract

We study analytically and numerically the stability of the standing waves for a nonlinear Schr\"odinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in $\hurad$ and unstable in $\hu$ under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a {\em finite-width instability}). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.