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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, Robert S. (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. doi:10.1007/s00220-008-0640-0 ISSN 0010-3616.
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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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Dates: |
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Volume: | Vol.284 | ||||
Number: | No.3 | ||||
Number of Pages: | 29 | ||||
Page Range: | pp. 803-831 | ||||
DOI: | 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) |
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