Weak Turbulence Theory for the Gross-Pitaevskii Equation

Abstract

Recent developments in the theory of Bose-Einstein condensates have resulted in a renewed interest in the Gross Pitaevskii equation (GPE),

$$ i\partial _t \psi + \Delta \psi - |\psi |^2 \psi - U\psi = 0, $$

((1))

where the potential U is a given function of coordinate, see for example Fig. 1. In the limit U = 0, the GPE is called the defocusing Nonlinear Schrödinger equation (NLSE), an equation which plays a central role in the understanding of non-linear optics.