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D. Pelinovsky

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Asymptotic reductions of the Gross-Pitaevskii equation

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Emergent Nonlinear Phenomena in Bose-Einstein Condensates,

Eds. P.G. Kevrekidis, D.J. Franzeskakis, and R. Carretero-Gonzalez,

Springer-Verlag, New York, pp. 377-398 (2008)

**Abstract:**

Various analytical techniques are reviewed in the context of asymptotic
reductions of the GrossPitaevskii (GP) equation, which is the nonlinear
Schršodinger (NLS) equation with an external potential. When the external
potential is periodic, the GP equation can be reduced to the coupled-mode
(Dirac) system, the continuous NLS equation and the discrete NLS equation
by using formal multi-scale expansion methods and their rigorous mathematical
analogues. When the external potential is decaying at infinity, finitedimensional
reductions of the GP equation can be derived for modeling of
dynamics of localized modes. When the external potential is confining, the GP
equation can be recovered from the multi-particle linear Schršodinger equation.

**Keywords**:

Gross-Pitaevskii equation, nonlinear Schrodinger equation, nonlinear Dirac equations,
discrete nonlinear Schrodinger equations, asymptotic methods