Nonlinear waves in random lattices: localization and spreading

Heterogeneity in lattice potentials (like random or quasiperiodic) can localize linear, non-interacting waves and halt their propagation. Nonlinearity induces wave interactions, enabling energy exchange and leading to chaotic dynamics. Understanding the interplay between the two is one of the topical problems of modern wave physics. In particular, one questions whether nonlinearity destroys localization and revives wave propagation, whether thresholds in wave energy/norm exist, and what the resulting wave transport mechanisms and characteristics are. Despite remarkable progress in the field, the answers to these questions remain controversial and no general agreement is currently achieved. This thesis aims at resolving some of the controversies in the framework of nonlinear dynamics and computational physics. Following common practice, basic lattice models (discrete Klein-Gordon and nonlinear Schroedinger equations) were chosen to study the problem analytically and numerically. In the disordered linear case all eigenstates of such lattices are spatially localized manifesting Anderson localization, while nonlinearity couples them, enabling energy exchange and chaotic dynamics. For the first time we present a comprehensive picture of different subdiffusive spreading regimes and self-trapping phenomena, explain the underlying mechanisms and derive precise asymptotics of spreading. Moreover, we have successfully generalized the theory to models with spatially inhomogeneous nonlinearity, quasiperiodic potentials, higher lattice dimensions and arbitrary nonlinearity index. Furthermore, we have revealed a remarkable similarity to the evolution of wave packets in the nonlinear diffusion equation. Finally, we have studied the limits of strong disorder and small nonlinearities to discover the probabilistic nature of Anderson localization in nonlinear disordered systems, demonstrating the finite probability of its destruction for arbitrarily small nonlinearity and exponentially small probability of its survival above a certain threshold in energy. Our findings give a new dimension to the theory of wave packet spreading in localizing environments, explain existing experimental results on matter and light waves dynamics in disordered and quasiperiodic lattice potentials, and offer experimentally testable predictions