Orbital stability and uniqueness of the ground state for the non-linear schrödinger equation in dimension one

Daniele Garrisi; Vladimir Georgiev

doi:10.3934/dcds.2017184

Orbital stability and uniqueness of the ground state for the non-linear schrödinger equation in dimension one

Daniele Garrisi^*
, Vladimir Georgiev

^*Corresponding author for this work

Research output: Journal Publication › Article › peer-review

7 Citations (Scopus)

Abstract

We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

Original language	English
Pages (from-to)	4309-4328
Number of pages	20
Journal	Discrete and Continuous Dynamical Systems
Volume	37
Issue number	8
DOIs	https://doi.org/10.3934/dcds.2017184
Publication status	Published - Aug 2017
Externally published	Yes

Free Keywords

Schrödinger
Stability
uniqueness

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2017184

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title = "Orbital stability and uniqueness of the ground state for the non-linear schr{\"o}dinger equation in dimension one",

abstract = "We prove that standing-waves which are solutions to the non-linear Schr{\"o}dinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.",

keywords = "Schr{\"o}dinger, Stability, uniqueness",

author = "Daniele Garrisi and Vladimir Georgiev",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary: 35Q55; Secondary: 47J35. Key words and phrases. Stability, uniqueness, Schr{\"o}dinger. The first author was supported by INHA UNIVERSITY Research Grant through the project number 51747-01 titled “Stability in non-linear evolution equations”. The second author was supported by University of Pisa, project no. PRA-2016-41 “Fenomeni singolari in problemi de-terministici e stocastici ed applicazioni”; by INDAM, GNAMPA - Gruppo Nazionale per l{\textquoteright}Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni and by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University. ∗ Corresponding author: Daniele Garrisi.",

year = "2017",

month = aug,

doi = "10.3934/dcds.2017184",

language = "English",

volume = "37",

pages = "4309--4328",

journal = "Discrete and Continuous Dynamical Systems",

issn = "1078-0947",

publisher = "American Institute of Mathematical Sciences",

number = "8",

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TY - JOUR

T1 - Orbital stability and uniqueness of the ground state for the non-linear schrödinger equation in dimension one

AU - Garrisi, Daniele

AU - Georgiev, Vladimir

N1 - Funding Information: 2010 Mathematics Subject Classification. Primary: 35Q55; Secondary: 47J35. Key words and phrases. Stability, uniqueness, Schrödinger. The first author was supported by INHA UNIVERSITY Research Grant through the project number 51747-01 titled “Stability in non-linear evolution equations”. The second author was supported by University of Pisa, project no. PRA-2016-41 “Fenomeni singolari in problemi de-terministici e stocastici ed applicazioni”; by INDAM, GNAMPA - Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni and by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University. ∗ Corresponding author: Daniele Garrisi.

PY - 2017/8

Y1 - 2017/8

N2 - We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

AB - We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

KW - Schrödinger

KW - Stability

KW - uniqueness

UR - https://www.scopus.com/pages/publications/85038121727

U2 - 10.3934/dcds.2017184

DO - 10.3934/dcds.2017184

M3 - Article

AN - SCOPUS:85038121727

SN - 1078-0947

VL - 37

SP - 4309

EP - 4328

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 8

ER -

Orbital stability and uniqueness of the ground state for the non-linear schrödinger equation in dimension one

Abstract

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