Asymptotic Reductions of the Gross-Pitaevskii Equation

Various analytical techniques are reviewed in the context of asymptotic reductions of the Gross—Pitaevskii (GP) equation, which is the nonlinear Schrödinger (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, finite-dimensional 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 Schrödinger equation.