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

Noriyoshi Fukaya; Vladimir Georgiev; Masahiro Ikeda

doi:10.1016/j.jde.2022.03.008

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

Noriyoshi Fukaya^*, Vladimir Georgiev, Masahiro Ikeda

^*Corresponding author for this work

Graduate School of Advanced Science and Engineering

Research output: Contribution to journal › Article › peer-review

103 Citations (Scopus)

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 (−Δ_α)_α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δ_α always has a unique simple negative eigenvalue e_α. We prove that if the frequency of the standing wave is close to −e_α, 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.

Original language	English
Pages (from-to)	258-295
Number of pages	38
Journal	Journal of Differential Equations
Volume	321
DOIs	https://doi.org/10.1016/j.jde.2022.03.008
Publication status	Published - 2022 Jun 5

Keywords

Instability
Nonlinear Schrödinger equation
Point interaction
Stability
Standing wave

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2022.03.008

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title = "On stability and instability of standing waves for 2d-nonlinear Schr{\"o}dinger equations with point interaction",

abstract = "We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schr{\"o}dinger equation with a point interaction and a focusing power nonlinearity. The Schr{\"o}dinger operator with a point interaction (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, 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.",

keywords = "Instability, Nonlinear Schr{\"o}dinger equation, Point interaction, Stability, Standing wave",

author = "Noriyoshi Fukaya and Vladimir Georgiev and Masahiro Ikeda",

note = "Funding Information: NF was supported by JSPS KAKENHI Grant Number JP20K14349 . VG was partially supported by Project {\textquoteleft}Problemi stazionari e di evoluzione nelle equazioni di campo non-lineari dispersive” of GNAMPA – Gruppo Nazionale per l'Analisi Matematica 2020, by the project PRIN 2020XB3EFL by the Italian Ministry of Universities and Research , by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa . MI is supported by JST CREST Grant Number JPMJCR1913 and by JSPS KAKENHI Grant Number JP19K14581 . Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

month = jun,

day = "5",

doi = "10.1016/j.jde.2022.03.008",

language = "English",

volume = "321",

pages = "258--295",

journal = "Journal of Differential Equations",

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T1 - On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

AU - Fukaya, Noriyoshi

AU - Georgiev, Vladimir

AU - Ikeda, Masahiro

N1 - Funding Information: NF was supported by JSPS KAKENHI Grant Number JP20K14349 . VG was partially supported by Project ‘Problemi stazionari e di evoluzione nelle equazioni di campo non-lineari dispersive” of GNAMPA – Gruppo Nazionale per l'Analisi Matematica 2020, by the project PRIN 2020XB3EFL by the Italian Ministry of Universities and Research , by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa . MI is supported by JST CREST Grant Number JPMJCR1913 and by JSPS KAKENHI Grant Number JP19K14581 . Publisher Copyright: © 2022 Elsevier Inc.

PY - 2022/6/5

Y1 - 2022/6/5

N2 - 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 (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, 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.

AB - 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 (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, 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.

KW - Instability

KW - Nonlinear Schrödinger equation

KW - Point interaction

KW - Stability

KW - Standing wave

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DO - 10.1016/j.jde.2022.03.008

M3 - Article

AN - SCOPUS:85126563092

SN - 0022-0396

VL - 321

SP - 258

EP - 295

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

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

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Keywords

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