PROPAGATION AND SCATTERING OF NONLINEAR-WAVES IN DISORDERED-SYSTEMS

We review recent numerical and analytical results related to the study of the coexistence of nonlinear wave propagation and disorder effects, mostly for soliton bearing systems. First of all, we briefly summarize scattering problems in the linear approximation which show many features related to the Anderson localization. Considering wavepackets instead of standard problems of the plane wave propagation, we show that the change of the asymptotic behaviour of the averaged transmission coefficient as a function of the disordered segment length may be demonstrated even at the level of the linear approach. For nonlinear problems we begin by discussing the stationary case which may be analyzed in detail. The main effect of nonlinearity, bistability, may be drastically modified by disorder in this case. However, for nonstationary problems disorder may give rise to more intensive development of nonlinear properties of physical systems via modulational instability, which is enhanced by inhomogeneities. We point out that localization effects created by disorder may vanish in the presence of strong nonlinearity, and we present examples showing that nonlinearity leads to an actual improvement of the transmission when it contributes to create soliton pulses. We analyze soliton propagation through disordered and inhomogeneous media and demonstrate that nonlinear transmission strongly depends on the soliton type: results are different for dynamical, topological, and envelope solitons.