Instability of bound states of nonlinear Schrödinger equations with Morse index equal to two

Masaya Maeda

doi:10.1016/j.na.2009.10.010

Instability of bound states of nonlinear Schrödinger equations with Morse index equal to two

Masaya Maeda

SCI - Mathematics

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

7 Citations (Scopus)

Abstract

We prove that every bound state of the nonlinear Schrödinger equation (NLS) with Morse index equal to two, with frac(d², d ω²) (E (φ{symbol}_ω) + ω Q (φ{symbol}_ω)) > 0, is orbitally unstable. We apply this result to two particular cases. One is the NLS equation with potential and the other is a system of three coupled NLS equations. In both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high.

Original language	English
Pages (from-to)	2100-2113
Number of pages	14
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	72
Issue number	3-4
DOIs	https://doi.org/10.1016/j.na.2009.10.010
Publication status	Published - 2010 Feb 1

Keywords

Bound states
Nonlinear Schrödinger equation
Orbital stability

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2009.10.010

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@article{24aca656cdfa4a9489d511ee48f79b0a,

title = "Instability of bound states of nonlinear Schr{\"o}dinger equations with Morse index equal to two",

abstract = "We prove that every bound state of the nonlinear Schr{\"o}dinger equation (NLS) with Morse index equal to two, with frac(d2, d ω2) (E (φ{symbol}ω) + ω Q (φ{symbol}ω)) > 0, is orbitally unstable. We apply this result to two particular cases. One is the NLS equation with potential and the other is a system of three coupled NLS equations. In both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high.",

keywords = "Bound states, Nonlinear Schr{\"o}dinger equation, Orbital stability",

author = "Masaya Maeda",

note = "Funding Information: The author would like to express his deep gratitude to Reika Fukuizumi for her helpful advice and discussions. He also wishes to express his sincere appreciation to Professor Yoshio Tsutsumi for his helpful advice and encouragement. The author was supported by Grant-in-Aid for JSPS Fellows (20 ⋅ 56371) from Japan Society for the Promotion of Science. The author is grateful to the referee for reading the manuscript of this paper carefully and for offering valuable suggestions. ",

year = "2010",

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doi = "10.1016/j.na.2009.10.010",

language = "English",

volume = "72",

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journal = "Nonlinear Analysis, Theory, Methods and Applications",

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number = "3-4",

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T1 - Instability of bound states of nonlinear Schrödinger equations with Morse index equal to two

AU - Maeda, Masaya

N1 - Funding Information: The author would like to express his deep gratitude to Reika Fukuizumi for her helpful advice and discussions. He also wishes to express his sincere appreciation to Professor Yoshio Tsutsumi for his helpful advice and encouragement. The author was supported by Grant-in-Aid for JSPS Fellows (20 ⋅ 56371) from Japan Society for the Promotion of Science. The author is grateful to the referee for reading the manuscript of this paper carefully and for offering valuable suggestions.

PY - 2010/2/1

Y1 - 2010/2/1

N2 - We prove that every bound state of the nonlinear Schrödinger equation (NLS) with Morse index equal to two, with frac(d2, d ω2) (E (φ{symbol}ω) + ω Q (φ{symbol}ω)) > 0, is orbitally unstable. We apply this result to two particular cases. One is the NLS equation with potential and the other is a system of three coupled NLS equations. In both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high.

AB - We prove that every bound state of the nonlinear Schrödinger equation (NLS) with Morse index equal to two, with frac(d2, d ω2) (E (φ{symbol}ω) + ω Q (φ{symbol}ω)) > 0, is orbitally unstable. We apply this result to two particular cases. One is the NLS equation with potential and the other is a system of three coupled NLS equations. In both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high.

KW - Bound states

KW - Nonlinear Schrödinger equation

KW - Orbital stability

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JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

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