Propagation of stationary nonlinear waves in disordered media B. Spivak A. Zyuzin

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 D=2,3 the absence of disorder.

STATIONARY NONLINEAR SCHRODINGER EQUATION. u(r) is a white noise scattering potential; l >>k-1 is the elastic mean free path.

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.

How many solutions f(r) does the problem have? 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 wave’s incidence dq, the scattering potential du(r) , and the wave’s frequency de ? c. Are these solutions stable?

Results for stationary nonlinear Schrodinger equation (D=3): The number of solutions of the nonlinear stationary Schoedinger equation increases exponentially with the sample size L, independently of the sign of b !

Stability of uniform waves in pure case (u(r)=0) in D=2,3 ? At b > 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 < 0 propagation of uniform waves is stable.

A review of results for linear diffusive case L>>1 . The average density and current density can be described by the diffusion equation (L>>l>>l). D= lv/3 is the diffusion coefficient. 2. The correlation function of the density fluctuations (D=3).

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 D=2 dN~1.

How to prove all of this? u(r) r r’ = P(r, r’) u’(r) G(r, r’) <u(r)u(r’)> u’(r) = u(r) + d u(r)

A warm up problem

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.

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!

The self-consistent equation

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 !!

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.

An example: I =2 g 2-2/3 g

Sensitivity of the solutions to changes of external parameters

An analogy with the spin glass problem

Stability of stationary solutions. Time dependent equations for “hydrodynamic variables” ui (t):

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 g>>1 the fraction of stable stationary solution is of order The number of stable stationary solutions is still exponentially large! c. At g>1 , depending on realization of u( r ), or F( r ), additionally one can have solutions which oscillate in time as well as strange attractors.

Ballistic case At large enough n0 there is no difference between integrable and nonintegrable geometries!

Strong non-linearity regime (pure case) It this case one can neglect the elastic mean free path l. Are uniform solutions stable at b > 0?

At I >> 1 the question is: If I=1 there are only stable and unstable fixes points. 2. If I=2 additionally one has limit cycles. They, however, have measure zero. 3. If I=3, additionally, one has strange attractors and chaos. At I >> 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?

Conclusions: Disordered 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.