Absence of Diffusion in a Nonlinear Schrödinger Equation

with a Random PotentialShmuel Fishman, Avy Soffer and

Yevgeny Krivolapov

The Equation2

0it

H

0 1 1n n n n nV H

,nV V x are random Anderson Model0H

1D lattice version

2

0 2

1( ) ( ) ( ) ( )

2x x V x x

x

H

1D continuum version

Experimental Relevance

• Nonlinear Optics

• Bose Einstein Condensates, aka Gross-Pitaevskii (GP) equation.

0 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.

• 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)

1/ n

2/ n

Does Dynamical localization survive nonlinearity ?

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).

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

2log x 2log x

t

st

Pikovsky, Shepelyansky

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

Our result

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

2t

Perturbation Theory

The nonlinear Schrödinger Equation on a Lattice in 1D

2

0n n n nit

H

1 1n n n n n 0H

n random Anderson Model0H

Eigenstates

( ) ( ) miE t mn m n

m

t c t e u m mn m nu E u0H

Enumeration of eigenstates

Anderson model eigenstates

| | | |( ) n nx x xnu x De

where is the localization center andnx 1/

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.

nx

nn x* F. Nakano, J. Stat. Phys. 123, 803 (2006)

21 2 3 1 3

1 2 3

1 2 3

( ), , *

, ,

m m m mi E E E E tm m mm m m m m

m m m

i c V c c c et

Overlap

1 2 3 31 2, ,m m m mm mmm n n n n

n

V u u u uof the range of the localization length

Perturbation expansion

0 1 11( ) ...... NN Nn n n n nc t c c c Q

Iterative calculation of lnc

is a remainder of the expansion nQ

that start at (0)

0( 0)n n nc c t

start at (0)

0n nc Example: The first order

01 ( )0000t

ni E E tnc iV e

0( )(1) 000

0

1 ni E E t

n nn

ec V

E E

0n 1 0000 0c iV t

The first problem: Secular terms

0n

The second problem: Small denominator problem

Elimination of secular terms

0 (1) 2 (2) ......n n n n nE E E E E

21 2 3 1 3

1 2 3

1 2 3

( ), , *

, ,

m m m mi E E E E tm m mm m m m m m m

m m m

i c E c V c c c et

For example:

0

0

1 0 1 1 0 ( )0000

0 1 1 ( )0000

n

n

i E E tt n n n n n

i E E tn n n n n

i c E c E c V e

E c E V e

0 1 0000 0 00nE E V

The problem of small denominators

0( )(1) 000

0

1 ni E E t

n nn

ec V

E E

example

,

0

1ss

n

DE E

0 1s

Fractional moments approach (Aizenman - Molchanov)

000( )

0

Pr n s nn

n

VKe e

E E

0 arbitrary

000000

0 0

1sup

ss nn

n sn n

VV Ke

E E E E

Localized eigenstatesChebychev

Bounding the remainder: Lowest order 0

0( )n n n n nc t Q E E '

0

1 2 1 2

1 2 1 2 3

1 2 1 2 3

( )000 00 { }

0 02 { } 3 { }

, , ,

ni E E t m i E tt n n n m

m

m m m mi E t i E tn m m n m m m

m m m m m

i Q V e V e Q

V e Q Q V e Q Q Q

{ }Ewhere indicates a generic sum of energies.

Integrating and taking the absolute value gives

Using the assumption| |n

nQ Me

2 2 2 3 30 1 2 3M G GM t G M t G M t

1 2

1 2

1 2

1 2

1 2 3

1 2 3

000000 2

,0

03

, ,

m mmnn n m n m m

m m mn

m mn m m m

m m m

VQ t V Q t V Q Q

E E

t V Q Q Q

Setting validates the bootstrap assumption.

and integrating by parts we get the bootstrap equation

2t

Main Result

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

20t

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.

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 ? st