Nonlinear Lattice Waves in Random Potentials

Abstract

Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transition, quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field. We will discuss recent advances in the dynamics of nonlinear lattice waves in random potentials. In the absence of nonlinear terms in the wave equations, Anderson localization is leading to a halt of wave packet spreading. Nonlinearity couples localized eigenstates and, potentially, enables spreading and destruction of Anderson localization due to nonintegrability, chaos and decoherence. The spreading process is characterized by universal subdiffusive laws due to nonlinear diffusion. We review extensive computational studies for one-and two-dimensional systems with tunable nonlinearity power. We also briefly discuss extensions to other cases where the linear wave equation features localization: Aubry-Andre localization with quasiperiodic potentials, Wannier-Stark localization with dc fields, and dynamical localization in momentum space with kicked rotors.

Superdiffusive transport and energy localization in disordered granular crystals

“…Some of the most prominent recent studies on these topics [1][2][3][4][5][6][7][8] have generalized to weakly nonlinear settings the ideas of P. W. Anderson, who showed theoretically that the diffusion of waves is curtailed in linear random media (where the randomness arises from defects or impurities) [9,10]. This interplay between disorder and nonlinearity -which often arises in the presence of lattice discreteness -is of considerable interest to a vast array of ongoing studies, as is evidenced by the recent reviews [11,12] (see also the numerous references therein). The set of different physical scenarios in which Anderson localization has been investigated is staggering: it ranges all the way from electromagnetism [1] and acoustics [7] to subjects such as quantum chromodynamics [13].…”

“…As in the above studies, we are interested in waves in disordered media, but we depart from the earlier work in a very important way: we seek to explore order-disorder transitions with a particular emphasis on strongly nonlinear media. This contrasts sharply with the linear and weakly nonlinear media in which Anderson-like models have traditionally been studied [11,12]. Our approach is motivated predominantly by the strong (and increasing) interest in granular crystals [14][15][16], which (as we discuss below) are very important both for the study of fundamental nonlinear phenomena and for numerous engineering applications.…”

“…This also makes them very wellsuited for investigating the effects of structural and material heterogeneities on nonlinear wave dynamics. Studies have examined the role of defects [22][23][24][25][26] (including in experimental settings [27,28]), interfaces between two arXiv:1411.5746v2 [cond-mat.dis-nn] 11 Sep 2015 different types of particles [29,30], decorated and/or tapered chains [31,32], chains of diatomic and triatomic units [33][34][35][36][37][38][39][40], and quasiperiodic and random configurations [41][42][43]. The tunability of granular crystals is valuable not only for fundamental studies of their underlying physics but also in potential engineering applicationsincluding shock and energy absorbing layers [30,43,44], sound focusing devices and delay lines [45], actuators [46], vibration absorption layers [35], sound scramblers [29,47], and acoustic switches and logic gates [48].…”

Scattering of waves by impurities in precompressed granular chains

“…Important themes in such studies have been transport properties of wavepackets and solitary waves and the interplay between disorder (especially in the context of Anderson localization), discreteness, and nonlinearity [28][29][30][31]. These themes are also relevant to a wide variety of other nonlinear lattice models [40,41].…”

Nanoptera in a Period-2 Toda Chain

“…Numerous studies have considered 'uniform' (i.e., monoatomic) one-dimensional (1D) lattices, which are chains in which each particle in the lattice is identical. However, particles need not be identical, and examinations of such 'heterogenous' lattices [51], with either periodic [43,44,65,79] or random [32,54] distributions of different particles, reveal a wealth of fascinating dynamics that do not arise in uniform lattices. Such dynamics include new families of solitary waves that have been observed in diatomic granular chains and which can exist only for discrete values of the ratio between the masses of the two particles in a diatomic unit [43].…”