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

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

3. localization
Does Localization Survive the Nonlinearity???

4. 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

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

6. Experimental Relevance
Nonlinear Optics
Bose Einstein Condensates (BECs)

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

13. Pikovsky, Sheplyansky

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

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

16. 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

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

18. start at
step 1
Secular term to be removed by
here

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

20. 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

21. Strategy for calculation of leading order
Equation for remainder term
Small denominators
Average
Probabilistic bound
Bootstrap

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

23. Where the equation for is

24. Where the linear part is
removed secular term

25. To simplify the calculations we will denote
such that

26. which leads to
Small denominators

27. Where Si are small divisor sums, defined bellow.

28. Small denominators

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

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

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

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

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

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

35. Using

38. Chebychev inequality
Therefore

39. same for
Therefore
similar for

40. Where the equation for is
Nonlinear terms
linear

41. The Bootstrap Argument

42. some algebra gives
or by renaming the constants

43. 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

44. Higher orders iterate number of products of in places
subtract forbidden combinations

45. 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

46. at order
exponential localization for

47. What is the effect of the remainder??

48. 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?