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

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2017, Volume 37, Issue 8: 4309-4328. Doi: 10.3934/dcds.2017184

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

Daniele Garrisi^1, , and
Vladimir Georgiev^2,3,

1.
West Building, Office No. 5W443, Department of Mathematics Education, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon, 402-751, South Korea
2.
Dipartimento di Matematica, Largo Bruno Pontecorvo n. 5, 56127, Pisa (PI), Italy
3.
Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

^× Corresponding author: Daniele Garrisi
^× Corresponding author: Daniele Garrisi

Received: November 2016

Revised: May 2017

Published: May 2017

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 deterministici 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.

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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.

Keywords:

Stability,
uniqueness,
Schrödinger.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 47J35.

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References

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