### Mathematics in Engineering

2021, Issue 2: 1-15. doi: 10.3934/mine.2021011
Research article Special Issues

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

• Received: 10 January 2020 Accepted: 27 June 2020 Published: 13 July 2020
• MSC : 35Q40, 35Q55

• In this paper we will continue the analysis of two dimensional Schr?dinger 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.

Citation: Riccardo Adami, Raffaele Carlone, Michele Correggi, Lorenzo Tentarelli. Stability of the standing waves of the concentrated NLSE in dimension two[J]. Mathematics in Engineering, 2021, 3(2): 1-15. doi: 10.3934/mine.2021011

### Related Papers:

• In this paper we will continue the analysis of two dimensional Schr?dinger 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.

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