Properties of Normal Modes in a Modified Disordered Klein-Gordon Lattice: From Disorder to Order

Abstract

We introduce a modified version of the disordered Klein-Gordon lattice model, having two parameters for controlling the disorder strength: D, which determines the range of the coefficients of the on-site potentials, and W, which defines the strength of the nearestneighbor interactions. We fix W = 4 and investigate how the properties of the system's normal modes change as we approach its ordered version, i.e. D → 0. We show that the probability density distribution of the normal mode's frequencies takes a 'U'-shaped profile as D decreases. Furthermore, we use two quantities for estimating the mode's spatial extent, the so-called localization volume V (which is related to the mode's second moment) and the mode's participation number P. We show that both quantities scale as ∝ D−2 when D approaches zero and we numerically verify a proportionality relation between them as V/P ≈ 2.6.

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

“…A single site excitation results to the excitation of several NMs, whose nonlinear interaction is responsible for the chaotic behavior of the wave-packet. As the spatial extent of the, nevertheless localized, NMs increases when W decreases [2,35,62], more NMs are excited by single site excitations for W = 4 than for W = 6. On top of that, the wider extent of these NMs lead to stronger interactions between them and in turn, to higher level of chaos as is observed in Fig.…”

“…In the presence of nonlinearity the dynamics becomes more complicated as the system's normal modes (NMs) couple and chaos appears. Thus, the interplay of disorder and nonlinearity has attracted extensive attention in theory [12,13,14,15,16,17,18,19,20,21,22,23,24,25], numerical simulations [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,…”