Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

Reika Fukuizumi; Andrea Sacchetti

doi:10.1007/s10955-014-1023-x

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

Reika Fukuizumi, Andrea Sacchetti

INF - System Information Sciences

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

9 Citations (Scopus)

Abstract

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is also discussed.

Original language	English
Pages (from-to)	707-738
Number of pages	32
Journal	Journal of Statistical Physics
Volume	156
Issue number	4
DOIs	https://doi.org/10.1007/s10955-014-1023-x
Publication status	Published - 2014 Jul

Keywords

Bose-Einstein condensates in periodic lattices
Nonlinear Schrödinger and Discrete Nonlinear Schrödinger equations
Semiclassical approximation

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s10955-014-1023-x

Cite this

@article{4c368fdefe8d4d109eb829e80dc9542a,

title = "Stationary States for Nonlinear Schr{\"o}dinger Equations with Periodic Potentials",

abstract = "In this paper we consider a one-dimensional non-linear Schr{\"o}dinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schr{\"o}dinger equation to a discrete non-linear Schr{\"o}dinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is also discussed.",

keywords = "Bose-Einstein condensates in periodic lattices, Nonlinear Schr{\"o}dinger and Discrete Nonlinear Schr{\"o}dinger equations, Semiclassical approximation",

author = "Reika Fukuizumi and Andrea Sacchetti",

note = "Funding Information: Acknowledgments The authors thank the referee for reading carefully the manuscript and giving many helpful comments to improve the demonstration. One of us (A.S.) is grateful for the hospitality of the Tohoku University where part of this paper was written. This work is supported by JSPS KAKENHI Grant Number 24740079. A.S. is partially supported by GNFM-INdAM.",

year = "2014",

month = jul,

doi = "10.1007/s10955-014-1023-x",

language = "English",

volume = "156",

pages = "707--738",

journal = "Journal of Statistical Physics",

issn = "0022-4715",

publisher = "Springer New York",

number = "4",

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T1 - Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

AU - Fukuizumi, Reika

AU - Sacchetti, Andrea

N1 - Funding Information: Acknowledgments The authors thank the referee for reading carefully the manuscript and giving many helpful comments to improve the demonstration. One of us (A.S.) is grateful for the hospitality of the Tohoku University where part of this paper was written. This work is supported by JSPS KAKENHI Grant Number 24740079. A.S. is partially supported by GNFM-INdAM.

PY - 2014/7

Y1 - 2014/7

N2 - In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is also discussed.

AB - In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is also discussed.

KW - Bose-Einstein condensates in periodic lattices

KW - Nonlinear Schrödinger and Discrete Nonlinear Schrödinger equations

KW - Semiclassical approximation

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U2 - 10.1007/s10955-014-1023-x

DO - 10.1007/s10955-014-1023-x

M3 - Article

AN - SCOPUS:84903701159

VL - 156

SP - 707

EP - 738

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

Abstract

Keywords

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