Universality in nonadiabatic behaviour of classical actions in nonlinear   models with separatrix crossings
  We discuss dynamics of approximate adiabatic invariants in several nonlinear models being related to physics of Bose-Einstein condensates (BEC). We show that nonadiabatic dynamics in Feshbach resonance passage, nonlinear Landau-Zener (NLZ) tunnelling, and BEC tunnelling oscillations in a double-well can be considered within a unifying approach based on the theory of separatrix crossings. The separatrix crossing theory was applied previously to some problems of classical mechanics, plasma physics and hydrodynamics, but has not been used in the rapidly growing BEC-related field yet. We derive explicit formulas for the change in the action in several models. Extensive numerical calculations support the theory and demonstrate its universal character. We also discovered a qualitatively new nonlinear phenomenon in a NLZ model which we propose to call {\em separated adiabatic tunnelling} 
A.P. Itin, S. Watanabe