Interactions destroy dynamical localization with strong and weak chaos

Bose-Einstein condensates loaded into kicked optical lattices can be treated as quantum kicked rotor systems. Noninteracting rotors show dynamical localization in momentum space. The experimentally tunable condensate interaction is included in a qualitative Gross-Pitaevskii type model based on two-body interactions. We observe strong and weak chaos regimes of wave packet spreading in momentum space. In the intermediate strong chaos regime the condensate energy grows as $t^{1/2}$. In the asymptotic weak chaos case the growth crosses over into a $t^{1/3}$ law. The results do not depend on the details of the kicking.