Propagation of stationary nonlinear waves in disordered media

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Propagation of stationary nonlinear waves in
disordered media

• B. Spivak
• A. Zyuzin

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• Plan
• Review of results on pure nonlinear Schroedinger
  (NLS) equation
• in 3D.
• Review of results on disordered linear
  Schroedinger equation.
• Solution of disordered nonlinear Schroedinger
  equation.
• 4. Discussion of similarity to a spin glass.
• Stability of stationary solutions. Temporal
  nonlinear speckles.
• Discussion of solutions of nonlinear Schroedinger
  equation in
• in D2,3 the absence of disorder.

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STATIONARY NONLINEAR SCHRODINGER EQUATION.
u(r) is a white noise scattering potential l
gtgtk-1 is the elastic mean free path.
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• The nonlinear Schroedinger equation is relevant
• for example, to following problems
• Plasma physics.
• Propagation of nonlinear EM waves in disordered
• media ( Kerr nonlinearity).
• c. Propagation of nonlinear waves in shallow
  liquids.
• In some aspects it is also similar to the problem
• of disordered interacting electrons in
  metals.

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Since u(r) is a random, f (r) is a random sample
specific function as well.

• Questions
• How many solutions f(r) does the problem have?
• How sensitive are these solutions to changes of
• parameters of the system such as
• the angle of the waves incidence dq,
• the scattering potential du(r) ,
• and the waves frequency de ?
• c. Are these solutions stable?

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Results for stationary nonlinear Schrodinger
equation (D3)
The number of solutions of the nonlinear
stationary Schoedinger equation increases
exponentially with the sample size L,
independently of the sign of b !
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Stability of uniform waves in pure case (u(r)0)
in D2,3 ?
At b gt 0 and arbitrary n0 spatially uniform
waves are unstable due to self-focusing
phenomena. The characteristic length where this
takes place is of order
At b lt 0 propagation of uniform waves is stable.
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A review of results for linear diffusive case
Lgtgt1 .

• The average density and current density can be
  described
• by the diffusion equation (Lgtgtlgtgtl).

D lv/3 is the diffusion coefficient.
2. The correlation function of the density
fluctuations (D3).

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3. How much should the frequency de be changed to
change completely the fluctuations of the
density?
4. How much should the angle of incidence dq be
changed to change the density fluctuations?
5. How many impurities should be moved to change
the speckle pattern completely? For example
in D2 dN1.
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How to prove all of this?
u(r)
P(r, r)
r
r
u(r)
G(r, r)
ltu(r)u(r)gt
u(r) u(r) d u(r)
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A warm up problem
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Limit of applicability of weak non-linearity
approximation

• It is a requirement that the self-focusing length
  l(sf) is less than
• the mean free path l. In this case the
  system is similar to a system
• of randomly distributed focusing and
  defocusing lenses.
• The term bn(r) represents an additional
  scattering potential.
• This criterion corresponds to a requirement
  that the mean free
• path associated with this potential is
  less than l.

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Solution of the general problem
Where ni(r) are eigenfunction of the diffusion
equation
with appropriate boundary conditions.
Solution of Schroedinger equation at fixed ui
. is a linear problem!
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The self-consistent equation
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Propertries of the random functions Fi(uj)

• 1. They fluctuate near constant averages
• 2. They are uncorrelated!
• 3. Their characteristic period is of order
  unity.
• Their derivatives in different directions are
  uncorrelated
• as well !!

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Out of the set of the equations
only the first
are relevant! The number of solutions of
the NLS equation with disorder is of order of a
volume of hyperparallepiped with sides g,
2-2/3g, . . .1.
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An example I 2
g
2-2/3 g
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Sensitivity of the solutions to changes of
external parameters
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An analogy with the spin glass problem
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Stability of stationary solutions. Time
dependent equations for hydrodynamic
variables ui (t)
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Results for the non-stationary equations
a. Near the first instability point there are
only 3 stationary solutions. Two of them are
stable and one is unstable. b. At ggtgt1 the
fraction of stable stationary solution is of
order The number of stable stationary
solutions is still exponentially large! c. At
ggt1 , depending on realization of u( r ), or F(
r ), additionally one can have solutions
which oscillate in time as well as strange
attractors.
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Ballistic case
At large enough n0 there is no difference
between integrable and nonintegrable geometries!
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Strong non-linearity regime (pure case)
It this case one can neglect the elastic mean
free path l.
Are uniform solutions stable at b gt 0?
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• If I1 there are only stable and unstable fixes
  points.
• 2. If I2 additionally one has limit cycles.
  They, however,
• have measure zero.
• 3. If I3, additionally, one has strange
  attractors and chaos.

At I gtgt 1 the question is What fraction of the
phase space is attracted to the stable fixed
points? What fraction of the phase space is
attracted to the chaotic regions? Is the ratio a
universal number?
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Conclusions Disor
dered nonlinear Schroedinger equation has
exponentially large number of stable stationary
solutions at large L. These solutions exhibit
exponential sensitivity to changes of initial
conditions, realizations of the scattering
potential, and the beam incidence angle. The
problem is quite similar to the spin glass
problem.