Chaotic shock waves of a Bose-Einstein condensate

It is demonstrated that the well-known Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose-Einstein condensate (BEC) driven by the time-periodic harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is constructed, which describes the matter shock waves with chaotic or periodic amplitudes and phases. When the periodic driving is switched off and the number of condensed atoms is conserved, we obtained the exact stationary states and non-stationary states. The former contains the stable non-propagated shock wave, and in the latter the shock wave alternately collapses and grows for the harmonic trapping or propagates with exponentially increased shock-front speed for the antitrapping. It is revealed that existence of chaos play a role for suppressing the blast of matter wave. The results suggest a method for preparing the exponentially accelerated BEC shock waves or the stable stationary states.