Anderson Localization for the Nonlinear Schrödinger Equation (NLSE):

Results and Puzzles

Yevgeny Krivolapov, Avy Soffer, and SF

The Nonlinear Schroedinger (NLS) Equation

2

oit

H

1 1n n n n nV 0H

V random Anderson Model0H

1D lattice version

2

2

1( ) ( ) ( ) ( )

2x x V x x

x

0H

1D continuum version

Does Localization Survive the Nonlinearity???

0 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

n1/ n

2/ n

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

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

2log x 2log x

t

st

Perturbation Theory

The nonlinear Schroedinger Equation on a Lattice in 1D

2

oit

H

1 1n n n n n 0H

n random Anderson Model0HEigenstates E0

m mn m nH u u

( ) ( ) miE t mn m n

m

t c t e u

The states are indexed by their centers of localization

mu

Is a state localized near m

This is possible since nearly each state has a Localization center and in box size M approximately M states

21 2 3 1 3

1 2 3

1 2 3

( ), , *

, ,tm m m mi E E E E tm m m

m m m m mm m m

i c V c c c e

Overlap 1 2 3 31 2, ,m m m mm mm

m n n n nn

V u u u u

of the range of the localization length

perturbation expansion

( ) (1) 2 (2) 1 ( 1)( ) ...... ....o N Nn n n n nc t c c c c

Iterative calculation of ( )lncstart at

(0)0( 0)n n nc c t

start at (0)

0n nc step 1

0( )(1) 000

tni E E t

n nc iV e

0n0( )

(1) 000

0

1 ni E E t

n nn

ec V

E E

0n (1) 0000t 0c iV

Secular term to be removed by ' (1) ....n n n nE E E E

here (1) 0000nE V

Secular terms can be removed in all orders by

' (1) 2 (2) ......n n n n nE E E E E

0n0( )

(1) 000

0

1 ni E E t

n nn

ec V

E E

The problem of small denominators

The Aizenman-Molchanov approach

'

( ) ( ) miE t mn m n

m

t c t e u 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

0 2

1t

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

'

( ) ( ) niE t nx n x

n

t c t e u 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 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

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 between is 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

Using

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

0 2

1t

Higher ordersiterate

21 2 3 1 3

1 2 3

1 2 3

1 1 1( ), , ( )* ( ) ( )

, , 0 0 0

t

m m m m

(N)m

N N r N r si E E E E tm m m r s l

m m m mm m m r s l

i c

V c c c e

NR number of products of 'V s

1N 'm s in 3 3N places

3 3

1N

NR

N

subtract forbidden combinations

ln 27lim ln 2

4N

N

R

N

Number of terms grows only exponentially, not factorially!!No entropy problem

typical term in the expansion

2 NNN N NR T e T

The only problem NT

it contains the information on the position dependence n

complicated because of small denominators

Can be treated producing a probabilistic bound

21 2( ) /( )Pr b n n Na N a NN

nc e e e e

1 3a 2 1a

at order

N exponential localization for

*n n* 2n N

1a

b

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?