The present paper reviews some of the recent theoretical developments in the field of multiple wave scattering in nonlinear disordered media. To be specific, we consider optical waves and restrict ourselves to the case of Kerr nonlinearity. Assuming that the nonlinearity is weak, we derive the expressions for the angular correlation functions and the coherent backscattering cone in a nonlinear disordered medium (Sec. 4). In both transmission and reflection, the short-range angular correlation functions of intensity fluctuations for two waves with different amplitudes (A and A′ → 0) appear to be given by the same expressions [Eqs. (6) and (7). respectively] as the angular correlation functions for waves at two different frequencies (ω and ω′ = ω - Δω) in a linear medium, with Δω replaced by 2lc/(3ξ~2), where ξ is a new nonlinear characteristic length defined by Eq. (20). The coherent backseattering cone is not affected by the nonlinearity, as long as the nonlinear coefficient ε_2 in Eq. (1) is purely real. If ε_2 has an imaginary part (which corresponds to the nonlinear absorption), the line shape of the cone is given by the same expression as in an absorbing linear medium, where the linear macroscopic absorption length L_a should be replaced by the generalized absorption length L_a~(NL) defined by Eq. (22). For the nonlinearity strength exceeding a threshold p approx= 1 [with the bifurcation parameter p given by Eq. (29)], we predict a new phenomenon ― temporal instability of the multiple-scattering speckle pattern ― to take place (Sec. 5). The instability is clue to a combined effect of the nonlinear self-phase modulation and the distributed feedback mechanism provided by multiple scattering and should manifest itself in spontaneous fluctuations of the speckle pattern with time. Since the spontaneous dynamics of the speckle pattern is irreversible, the time-reversal symmetry is spontaneously broken when p surpasses 1. The important feature of our result is the extensive nature of the instability threshold, leading to an interesting possibility of obtaining unstable regimes even at very weak nonlinearities, provided that the disordered sample is large enough. To study the dynamics of multiple-scattering speckle patterns beyond the instability threshold, we generalize the Langevin description of wave diffusion in disordered media (Sec. 5.3). Explicit expressions for the characteristic time scale of spontaneous intensity fluctuations are derived with account for the noninstan-taneous nature of the nonlinearity. The results of this study allow us to hypothesize that the dynamics of the speckle pattern may become chaotic immediately beyond the instability threshold, and that the cascade of bifurcations, typical for chaotic transitions in many known nonlinear systems, might not be present in the considered case of diffuse waves.
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