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

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

localization Does Localization Survive the Nonlinearity???

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) ????

Pikovsky, Sheplyansky

Pikovsky, Shepelyansky S.Flach, D.Krimer and S.Skokos 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

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

start at step 1 Secular term to be removed by here

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

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 Small denominators Average Probabilistic bound Bootstrap

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

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 Si are small divisor sums, defined bellow.

Small denominators

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

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

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

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

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

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

Using

Chebychev inequality Therefore

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 in places 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 order exponential 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?