Parametric resonance of ground states in the nonlinear Schrödinger equation

S. Cuccagna; E. Kirr; D. Pelinovsky

doi:10.1016/j.jde.2005.07.009

Parametric resonance of ground states in the nonlinear Schrödinger equation

S. Cuccagna, E. Kirr, D. Pelinovsky

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

Abstract

We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t^-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.

Original language	English (US)
Pages (from-to)	85-120
Number of pages	36
Journal	Journal of Differential Equations
Volume	220
Issue number	1
DOIs	https://doi.org/10.1016/j.jde.2005.07.009
State	Published - Jan 1 2006
Externally published	Yes

Keywords

Asymptotic stability
Ground states
Nonlinear Schrödinger equation
Parametric resonance

ASJC Scopus subject areas

Analysis
Applied Mathematics

Online availability

10.1016/j.jde.2005.07.009

Library availability

Discover UIUC Full Text

Cite this

@article{8ec8f932510f420aba7ea7dfab347608,

title = "Parametric resonance of ground states in the nonlinear Schr{\"o}dinger equation",

abstract = "We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schr{\"o}dinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.",

keywords = "Asymptotic stability, Ground states, Nonlinear Schr{\"o}dinger equation, Parametric resonance",

author = "S. Cuccagna and E. Kirr and D. Pelinovsky",

note = "The authors thank M. Weinstein for discussions of the problem. The work was initiated by the program “Resonances in linear and nonlinear Schr{\"o}dinger equations” at Fields Institute, August 2003. E. Kirr was partially supported by NSF Grant DMS-0405921. S. Cuccagna was supported by a grant from the Italian government. D. Pelinovsky was partially supported by grants from NSERC and PREA.",

year = "2006",

month = jan,

day = "1",

doi = "10.1016/j.jde.2005.07.009",

language = "English (US)",

volume = "220",

pages = "85--120",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Parametric resonance of ground states in the nonlinear Schrödinger equation

AU - Cuccagna, S.

AU - Kirr, E.

AU - Pelinovsky, D.

N1 - The authors thank M. Weinstein for discussions of the problem. The work was initiated by the program “Resonances in linear and nonlinear Schrödinger equations” at Fields Institute, August 2003. E. Kirr was partially supported by NSF Grant DMS-0405921. S. Cuccagna was supported by a grant from the Italian government. D. Pelinovsky was partially supported by grants from NSERC and PREA.

PY - 2006/1/1

Y1 - 2006/1/1

N2 - We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.

AB - We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.

KW - Asymptotic stability

KW - Ground states

KW - Nonlinear Schrödinger equation

KW - Parametric resonance

UR - http://www.scopus.com/inward/record.url?scp=28344456247&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28344456247&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2005.07.009

DO - 10.1016/j.jde.2005.07.009

M3 - Article

AN - SCOPUS:28344456247

SN - 0022-0396

VL - 220

SP - 85

EP - 120

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -

Parametric resonance of ground states in the nonlinear Schrödinger equation

Abstract

Keywords

ASJC Scopus subject areas

Online availability

Library availability

Related links

Fingerprint

Cite this