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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. 803831. ISSN 00103616
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Official URL: http://dx.doi.org/10.1007/s0022000806400
Abstract
We justify the use of the lattice equation (the discrete nonlinear Schrodinger equation) for the tightbinding 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 bandgap spectrum and the FourierBloch 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 piecewiseconstant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a nonzero width. The discrete nonlinear Schrodinger equation is applied to classify localized solutions of the GrossPitaevskii 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:  00103616  
Official Date:  December 2008  
Dates: 


Volume:  Vol.284  
Number:  No.3  
Number of Pages:  29  
Page Range:  pp. 803831  
Identification Number:  10.1007/s0022000806400  
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 HumboldtStiftung (AvHS), Deutsche Forschungsgemeinschaft (DFG), BadenWürttemberg (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 
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