Alexander Itin, Shinichi Watanabe
         (University of Electro-Communications) 

         Title

         Nonlinear dynamics of Bose-Einstein condensates:
         nonlinear two-level model with slowly changing parameters
         and dynamical stabilization of matter-wave breathers.

         Abstract

         We consider two problems related to nonlinear dynamics of
         Bose-Einstein condensates (BEC).  The problem of dynamical
         stabilization of BEC by rapidly oscillating scattering
         length has been studied before by several methods:
         Gaussian variational approximation, the method of moments,
         method of modulated Townes soliton, and the direct
         averaging of the Gross-Pitaevskii equation. We reconsider
         these methods and find serious discrepancy between the
         numerically obtained stabilized solution and that assumed
         by the theoretical methods. Interesting new features of
         stabilized solutions are revealed.
         Secondly, we consider nonlinear two-level model with slowly
         changing parameters. This problem arise in such systems as
         BEC in two asymmetric wells coupled by tunelling,
          BEC in an accelerating optical lattice etc. The equations
         of motion form a particular example of a Hamiltonian system
         with separatrix crossings for which there is a rigorous
         averaging technique developed by  A.I. Neishtadt . It was
         applied previously to several problems in plasma physics,
         hydrodynamics, classical mechanics.  We apply it  to the
         nonlinear two-level problem with slowly changing parameters
         in the context of BEC dynamics.   Nonadiabatic quantum
         dynamics can be described in terms of jumps of
         corresponding classical action variables for which
         general formulas exist.