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 Jan 1

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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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.",

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AU - Sacchetti, Andrea

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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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