We consider the case of a cubic nonlinear Schr\"odinger equation with an additional chaotic potential, in the sense that such a potential produces chaotic dynamics in classical mechanics. We derive and describe an appropriate semiclassical limit to such a nonlinear Schr\"odinger equation, using a semiclassical interpretation of the Wigner function, and relate this to the hydrodynamic limit of the Gross-Pitaevskii equation used in the context of Bose-Einstein condensation. We investigate a specific example of a Gross-Pitaevskii equation with such a chaotic potential, the one-dimensional {δ}-kicked harmonic oscillator, and its semiclassical limit, discovering in the process an interesting interference effect, where increasing the strength of the repulsive nonlinearity promotes localization of the wave function. We explore the feasibility of an experimental realization of such a system in a Bose-Einstein condensate experiment, giving a concrete proposal of how to implement such a configuration, and considering the problem of condensate depletion.
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Gardiner et al. (2000) studied this question.