Invariant-imbedding approach to localization. II. Non-linear random media

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Abstract
By employing an invariant-imbedding method a partial differential equation is derived for the complex reflection amplitude R (L ) of a one-dimensional non-linear random medium of length L. The method of characteristics reduces this equation to a dynamical system. Averaging of the perturbation of orbits by weak disorder is used to investigate the probability distribution of R (L). Two different situations are considered : fixed output w 0 (Problem A) and fixed input (Problem B). For a large class of non-linearities the generic behaviour for Problem A is as follows : i) For weak non-linearities, a crossover between an exponential decay of transmission ∼ exp(- L/4 ξ) at short L and a power law decay at large L is shown to take place at a length scale L * = ξ In (1/w0), ii) Strong non-linearities dominate the full behaviour and give rise to a power law decay. The physical origin of this behaviour is traced back to the enhancement of non-linearities by disorder. For Problem B, the asymptotic behaviour is shown to be always an exponential decay. The fluctuations associated with both regimes are obtained. Random non-linearities are also investigated and shown to lead to a self-repelling phenomenon at finite distances. The relevance of our results to experimental situations is briefly discussed