Spreading in disordered lattices with different nonlinearities

We study the spreading of initially localized states in a nonlinear disordered lattice described by the nonlinear Schrödinger equation with random on-site potentials —a nonlinear generalization of the Anderson model of localization. We use a nonlinear diffusion equation to describe the subdiffusive spreading. To confirm the self-similar nature of the evolution we characterize the peak structure of the spreading states with help of Rényi entropies and in particular with the structural entropy. The latter is shown to remain constant over a wide range of time. Furthermore, we report on the dependence of the spreading exponents on the nonlinearity index in the generalized nonlinear Schrödinger disordered lattice, and show that these quantities are in accordance with previous theoretical estimates, based on assumptions of weak and very weak chaoticity of the dynamics. Copyright c © EPLA, 2010 In disordered 1D lattices, all eigenmodes are exponentially localized due to Anderson localization [1]. These models first appeared in the area of disordered electronic systems [2,3], but they are also applicable to a wide variety of phenomena in a general context of waves (optical, acoustical, etc.) in disordered media [4–6]. Localization effectively prevents spreading of energy in such situations. By considering waves of large amplitudes, one faces nonlinearity and naturally encounters the question whether the nonlinearity destroys the localization or not. Although this question has already been addressed numerically [7–12], experimentally in BECs [13–15] and optical waveguides [16,17] as well as mathematically [18], a full understanding is still elusive. It is easier to understand how nonlinearity destroys localization leading to thermalization [19] and self-transparency [20] in short random lattices, than to analyze asymptotic regimes at large times in long lattices. For the latter setups a similarity between the quantum kicked rotor and a 1D Anderson model [21–23] has provided an alternative realization of the effects of nonlinearity. In this paper, we study the structural properties of the spreading field in nonlinear disordered lattices, focusing on their dependence on the nonlinearity index. Indeed, initial studies of the spreading of perturbations [7,9,10] have been almost exclusively restricted to the behavior of the second moment of the distribution and of the participation number. These quantities, however, do not allow one to distinguish between all possible scenarios. In particular, the second moment of the distribution can grow due to a uniform spreading of the field, but also when localized packages move in opposite directions. Additionally, both these processes may coexist with some bursts that do not spread at all. In order to resolve these structural features in a statistical way, we apply for the first time a characterization of the spreading fields in nonlinear lattices with generalized Rényi entropies. For guidance, we compare the spreading properties with that of the nonlinear diffusion equation and study the relation between the effective diffusion index with the nonlinearity index of the original model. Our basic model is described by the following generalization of the Discrete Anderson Nonlinear Schrödinger Equation (gDANSE): i d dt ψn = Vnψn +ψn−1+ψn+1+β|ψn| ψn . (1) Here n= 1, . . . , N is the lattice site index and Vn is the uncorrelated random potential, chosen uniformly from the intervall [−W/2,W/2]. The coefficient β is proportional to the nonlinear strength (hereafter we assume a normalization ∑ n |ψn| 2 = 1). In this work, we consider only the case β = 1 and W = 4. The parameter α, which we call nonlinearity index, is a novelity compared to the standard DANSE model with α= 1 [9,10]. Without nonlinearity β = 0, eq. (1) is a standard Anderson model describing a