1. Shmuel Fishman, Avy Soffer and Yevgeny Krivolapov
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
Anderson Model

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

17. Example: The first order
start at
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. Localized eigenstates Chebychev
arbitrary

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

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 , 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 ?