  We consider the case of a cubic nonlinear Schr\"{o}dinger 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\"{o}dinger 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
delta-kicked harmonic oscillator, and its semiclassical limit. We explore the
feasibility of 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.
