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D. Pelinovsky, G. Schneider, and R. MacKay

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

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**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 by the Wannier functions and the problem
is reduced, with the elliptic theory, to a set of nonlinear
algebraic equations solvable with the Implicit Function Theorem.
Our analysis is valid 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.

**Keywords**:

justification of amplitude equations, gap solitons in periodic potentials,
Gross-Pitaevskii equation, Fourier-Bloch decomposition, Wannier functions,
discrete nonlinear Schrodinger equation