Stability of the standing waves of the concentrated NLSE in dimension two

Abstract

In this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

References in corpus (11)

• Normalized solutions for Nonlinear Schrödinger systems on bounded domains
• On Schroedinger Equations with Concentrated Nonlinearities
• Stability and symmetry-breaking bifurcation for the ground states of a NLS with a ${\delta }^{\prime }$ interaction
• Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three
• Blow-up for the pointwise NLS in dimension two: absence of critical power
• The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions
• Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer
• Well-posedness of the Two-dimensional Nonlinear Schrödinger Equation with Concentrated Nonlinearity
• Analytic theory of narrow lattice solitons
• Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
• The 1-D Dirac equation with concentrated nonlinearity

Cited by in corpus (9)

• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect
• Complete ionization for a non-autonomous point interaction model in d = 2
• A general review on the NLS equation with point-concentrated nonlinearity
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions