Absence of Diffusion in a Nonlinear Schrödinger Equation with a Random Potential

1. Absence of Diffusion in a Nonlinear Schrödinger Equation with a Random Potential Shmuel Fishman, Avy Soffer and Yevgeny Krivolapov

2. The Equation 1D lattice version 1D continuum version Anderson Model are random

3. Experimental Relevance • Nonlinear Optics • Bose Einstein Condensates, aka Gross-Pitaevskii (GP) equation.

7. What is known ? • Localization: At high disorder all the eigenstates of almost every realization of the disorder are exponentially localized. • Dynamical Localization: Any transfer or spreading of wavepackets is suppressed.

8. Does Dynamical localization survive nonlinearity ? • Yes, if there is spreading the magnitude of the nonlinear term decreases and localization takes over. • Depends, assume localization length is then the relevant energy spacing is , the perturbation because of the nonlinear term is and all depends on (Shepelyansky) • No, there will be spreading for every value of (Flach) • Yes, because quasiperiodic localized perturbation does not destroy localization (Soffer, Wang-Bourgain)

9. Numerical Simulations • In regimes relevant for experiments looks that localization takes place • Scattering results (Paul, Schlagheck, Leboeuf, Pavloff, Pikovsky) • Spreading for long time. Finite time-step integration, no convergence to true solution, due to Chaos effects (Shepelyansky, Pikovsky, Molina, Flach, Aubry).

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

11. Pikovsky, Shepelyansky

12. Problem of all numerics Asymptotics problem: It is impossible to decide whether there is a saturation in the expansion of the wavefunction. In any case it looks like the expansion is very slow, at most sub-diffusional. Convergence: All long time numerics are done without convergence to true solution. Time scale: The time scale of the problem is not clear

13. Our result • For times the wavepacket is exponentially localized, namely, no spreading takes place.

14. Perturbation Theory The nonlinear Schrödinger Equation on a Lattice in 1D random Anderson Model Eigenstates

15. Enumeration of eigenstates Anderson model eigenstates where is the localization center and Since it was proven* that there is a finite number of eigenstates for any finite box around we will enumerate the eigenstates using their localization center. * F. Nakano, J. Stat. Phys. 123, 803 (2006)

16. Overlap of the range of the localization length Perturbation expansion is a remainder of the expansion that start at Iterative calculation of

17. start at Example: The first order The first problem: Secular terms The second problem: Small denominator problem

18. Elimination of secular terms For example:

19. The problem of small denominators example Fractional moments approach (Aizenman - Molchanov)

20. arbitrary Localized eigenstates Chebychev

21. where indicates a generic sum of energies. Bounding the remainder: Lowest order

22. Integrating and taking the absolute value gives Using the assumption and integrating by parts we get the bootstrap equation Setting validates the bootstrap assumption.

23. Main Result • Starting from a localized state the wave function is exponentially bounded for time of the order of

24. The logarithmic spreading conjecture • Wei-Min Wang conjectured that the solutions do not spread faster than (possibly logarithmically) in a meeting in June 2008 (Technion). It was based on her paper (with Z. Zhang): "Long time Anderson localization for nonlinear random Schroedinger equation", ArXiv 0805.3520, which she presented during this meeting. She made a similar conjecture earlier in a private communication. • We conjectured that the spreading of solutions is at most like in meetings that took place in December 2007 in talks by S. Fishman (IHP/Paris) and Y. Krivolapov (Weizmann). It was based on work in progress assuming that the remainder term can be bounded.

25. Questions • How can one understand the numerical results? Are they transient ? • Is there a time-scale for cross-over, and what is its experimental relevance ? • Does localization survive sufficiently strong nonlinearity? • What is the physics of ?