Nonlinear wave propagation in a random media and delocalization of quantum chaos by weak nonlinearity

The manifestation of dynamical localization can take place also in the propagation of linear waves in wave guides or fibers (Fishman, Prange 1989). For propagating waves the localization suppresses the growth of aperture angle with the wave guide length and leads to effective intensity transmission. Here a new and interesting type of problem arises if the waves propagate in a nonlinear media. This problem puts the question of general interest: how the localization, appearing as the result of linear wave interference, is modified by the introduction of small nonlinear wave interaction? It is shown [62] that there is a critical strength of nonlinear coupling below which the localization remains. Above this border a delocalization takes place and the number of excited linear modes grows according to the derived anomalous subdiffusion law ( $(\Delta n)^2 \sim t^{2/5}$). This excitation is much slower than the chaotic diffusion of classical rays so that the suppression of classical chaos by quantum (or linear waves) interference is not completely destroyed. The obtained subdiffusion law is of universal nature since it always takes place in the limit of weak nonlinearity when the energy of nonlinear four-waves interaction ( ${\mid{\psi}\mid}^4$) is much less than the energy of linear modes. The obtained results [62,68,49] show that the penetration of nonlinear waves through a one-dimensional disordered media decays exponentially with the length of the layer if a constant of nonlinear interaction is less than some critical value. Above this threshold subdiffusional propagation through the layer takes place and after some time the wave crosses the layer without any significant loss of the amplitude. The presented picture is quite different from the picture of Souillard and Doucot and Rammal obtained in stationary approximation according to which there is no critical value of nonlinearity. These studies initiated further researches of two interacting particles in a random potential [69].