D. Pelinovsky and G. Schneider
Bounds on the tight-binding approximation for
the Gross-Pitaevskii equation with a periodic potential
Journal of Differential Equations 248, 837-849 (2010)
Abstract:
We justify the validity of the discrete nonlinear Schrodinger
equation for the tight-binding approximation in the context of the
Gross-Pitaevskii equation with a periodic potential. Both piecewise-constant and smooth
potentials are considered in the semi-classical limit. While justification of
stationary equations is developed in our previous work, this work
deals with time-dependent space-decaying solutions on large but finite
time intervals.
Keywords:
justification of amplitude equations, gap solitons in periodic potentials,
Gross-Pitaevskii equation, Fourier-Bloch decomposition, Wannier functions,
discrete nonlinear Schrodinger equation