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Anderson Localization for the Nonlinear Schrödinger Equation (NLSE): Results and Puzzles Yevgeny Krivolapov, Avy Soffer, and SF

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The Nonlinear Schroedinger (NLS) Equation random Anderson Model 1D lattice version 1D continuum version

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Does Localization Survive the Nonlinearity??? localization

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Does 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 is and all depends on No, but does not depend on No, but it depends on realizations Yes, because time dependent localized perturbation does not destroy localization

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Does Localization Survive the Nonlinearity??? Yes, wings remain bounded No, the NLSE is a chaotic dynamical system.

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Experimental Relevance Nonlinear Optics Bose Einstein Condensates (BECs)

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Numerical Simulations In regimes relevant for experiments looks that localization takes place Spreading for long time (Shepelansky, Pikovsky, Molina, Kopidakis, Komineas Flach, Aubry) ????

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Pikovsky, Sheplyansky

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S.Flach, D.Krimer and S.SkokosPikovsky, Shepelyansky t

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Perturbation Theory The nonlinear Schroedinger Equation on a Lattice in 1D random Anderson Model Eigenstates

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The states are indexed by their centers of localization Is a state localized near This is possible since nearly each state has a Localization center and in box size M approximately M states

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Overlap of the range of the localization length perturbation expansion Iterative calculation of start at

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step 1 Secular term to be removed by here

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Secular terms can be removed in all orders by The problem of small denominators The Aizenman-Molchanov approach New ansatz

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Leading order result Starting 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

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Strategy for calculation of leading order Equation for remainder term AverageSmall denominators Probabilistic bound Bootstrap

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Equation for remainder term We need to show or just We will limit ourselves to the expansion remainder Expansion

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Where the equation for is

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Where the linear part is removed secular term

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To simplify the calculations we will denote such that

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which leads to Small denominators

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Where S i are small divisor sums, defined bellow.

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Small denominators

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We split the integration to boxes with constant sign of the Jacobian Using the following inequality

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The integral can be bounded The integral could be further bounded where We show this in what follows

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Use Feynman-Hellman theorem and take j significantly far. Sub-exponential

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Minami estimate We exclude intervals Exclude realizations of a small measure the difference betweenis not small

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The effect of renormalization Using (leading order in ) One finds where are the minimal and maximal Lyapunov exponents Therefore is not effected for

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A bound on the overlap integral is Bound on the growth of the linear term

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Using

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Therefore Chebychev inequality

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same for Therefore similar for

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Where the equation for is Nonlinear terms linear

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The Bootstrap Argument

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some algebra gives or by renaming the constants

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Summary Starting 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

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Higher orders iterate number of products of inplaces subtract forbidden combinations

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Number of terms grows only exponentially, not factorially!! No entropy problem typical term in the expansion The only problem it contains the information on the position dependence complicated because of small denominators Can be treated producing a probabilistic bound

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at orderexponential localization for

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What is the effect of the remainder??

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Questions 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?

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