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### Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential

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Pelinovsky, Dmitry, Schneider, Guido and MacKay, R. S. (Robert Sinclair).
(2008)
*Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential.*
Communications in Mathematical Physics, Vol.284
(No.3).
pp. 803-831.
ISSN 0010-3616

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1007/s00220-008-0640-0

## Abstract

We justify the use of the lattice equation (the discrete nonlinear Schrodinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier-Bloch decomposition for a linear Schrodinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrodinger equation is applied to classify localized solutions of the Gross-Pitaevskii equation with a periodic potential.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Schrödinger equation, Nonlinear theories, Dynamics, Differential equations, Elliptic -- Numerical solutions |

Journal or Publication Title: | Communications in Mathematical Physics |

Publisher: | Springer |

ISSN: | 0010-3616 |

Official Date: | December 2008 |

Volume: | Vol.284 |

Number: | No.3 |

Number of Pages: | 29 |

Page Range: | pp. 803-831 |

Identification Number: | 10.1007/s00220-008-0640-0 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), Alexander von Humboldt-Stiftung (AvHS), Deutsche Forschungsgemeinschaft (DFG), Baden-Württemberg (Germany) |

Grant number: | EP/D069513/1 (EPSRC) |

References: | 1. Alfimov, G.L., Kevrekidis, P.G., Konotop, V.V., Salerno, M.:Wannier functions analysis of the nonlinear |

URI: | http://wrap.warwick.ac.uk/id/eprint/29127 |

Data sourced from Thomson Reuters' Web of Knowledge

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