3localizationDoes Localization Survive the Nonlinearity???
4Does Localization Survive the Nonlinearity??? Yes, if there is spreading the magnitude of the nonlinear term decreases and localization takes over.No, assume wave-packet width is then the relevant energy spacing is , the perturbation because of the nonlinear term isand all depends onNo, but does not depend onNo, but it depends on realizationsYes, because time dependent localized perturbation does not destroy localization
5Does Localization Survive the Nonlinearity??? Yes, wings remain boundedNo, the NLSE is a chaotic dynamical system.
6Experimental Relevance Nonlinear OpticsBose Einstein Condensates (BECs)
43SummaryStarting from a state localized at some site the wave function is exponentially small at distances larger than the localization length for time of the order
44Higher orders iterate number of products of in places subtract forbidden combinations
45Number of terms grows only exponentially, not factorially!! No entropy problemtypical term in the expansionThe only problemit contains the information on the position dependencecomplicated because of small denominatorsCan be treated producing a probabilistic bound
48Questions and Puzzles What is the long time asymptotic behavior?? How can one understand the numerical results??What is the time scale for the crossover to the long time asymptotics??Are there parameters where this scale is short?