Mathematics Colloquia and Seminars

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We consider the Gross-Pitaevskii equation (cubic NLS) in 3-d with a trapping but non-confining potential. The equation is the mean field model for Bose-Einstein condensates and, at least for small initial data the solution evolves into a nonlinear ground state (the condensate) and a dispersive part. In particular the ground state is asymptotically stable. Numerical simulation and averaging techniques predicted that if the coefficient in front of the nonlinearity becomes oscillatory in time, the ground state evolves into a quasi-periodic solution (breather like solitary wave). We rigorously show that this is not possible if the potential is not confining. Moreover, for any small initial data the solution disperses to infinity on a time scale that we determine. The former stable ground state is destroyed by a radiation damping mechanism induced by resonance with continuum spectrum. I will present in a detailed and self contained manner the mathematics behind this phenomenon. This is joint work with S. Cuccagna and D. Pelinovsky