On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

2021/09/10 by Fukaya, Noriyoshi, Georgiev, Vladimir, Ikeda, Masahiro · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.04680

Abstract

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger 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 L²-subcritical or critical case, while the instability in the L²-supercritical case.

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• Point interactions and singular solutions to semilinear elliptic equations

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