Presentation on theme: "Anderson Localization for the Nonlinear Schrödinger Equation (NLSE): Results and Puzzles Yevgeny Krivolapov, Avy Soffer, and SF."— Presentation transcript:
Anderson Localization for the Nonlinear Schrödinger Equation (NLSE): Results and Puzzles Yevgeny Krivolapov, Avy Soffer, and SF
The Nonlinear Schroedinger (NLS) Equation random Anderson Model 1D lattice version 1D continuum version
Does Localization Survive the Nonlinearity??? localization
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
Does Localization Survive the Nonlinearity??? Yes, wings remain bounded No, the NLSE is a chaotic dynamical system.
Experimental Relevance Nonlinear Optics Bose Einstein Condensates (BECs)
Numerical Simulations In regimes relevant for experiments looks that localization takes place Spreading for long time (Shepelansky, Pikovsky, Molina, Kopidakis, Komineas Flach, Aubry) ????
S.Flach, D.Krimer and S.SkokosPikovsky, Shepelyansky t
Perturbation Theory The nonlinear Schroedinger Equation on a Lattice in 1D random Anderson Model Eigenstates
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
Overlap of the range of the localization length perturbation expansion Iterative calculation of start at
step 1 Secular term to be removed by here
Secular terms can be removed in all orders by The problem of small denominators The Aizenman-Molchanov approach New ansatz
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
Strategy for calculation of leading order Equation for remainder term AverageSmall denominators Probabilistic bound Bootstrap
Equation for remainder term We need to show or just We will limit ourselves to the expansion remainder Expansion
Where the equation for is
Where the linear part is removed secular term
To simplify the calculations we will denote such that
which leads to Small denominators
Where S i are small divisor sums, defined bellow.
We split the integration to boxes with constant sign of the Jacobian Using the following inequality
The integral can be bounded The integral could be further bounded where We show this in what follows
Use Feynman-Hellman theorem and take j significantly far. Sub-exponential
Minami estimate We exclude intervals Exclude realizations of a small measure the difference betweenis not small
The effect of renormalization Using (leading order in ) One finds where are the minimal and maximal Lyapunov exponents Therefore is not effected for
A bound on the overlap integral is Bound on the growth of the linear term
Therefore Chebychev inequality
same for Therefore similar for
Where the equation for is Nonlinear terms linear
The Bootstrap Argument
some algebra gives or by renaming the constants
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
Higher orders iterate number of products of inplaces subtract forbidden combinations
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
at orderexponential localization for
What is the effect of the remainder??
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?