Existence and stability of standing waves for nonlinear fractional Schrödinger equations J. Math. Phys. 53, 083702 (2012) N-fold Darboux transformations and soliton solutions of three nonlinear equations J. Math. Phys. 53, 083502 (2012) Some algebro-geometric solutions for the coupled modified Kadomtsev-Petviashvili equations arising from the Neumann type systems
In the context of the analysis of measured data, one is often faced with the task to differentiate data numerically. Typically, this occurs when measured data are concerned or data are evaluated numerically during the evolution of partial or ordinary differential equations. Usually, one does not take care for accuracy of the resulting estimates of derivatives because modern computers are assumed to be accurate to many digits. But measurements yield intrinsic errors, which are often much less accurate than the limit of the machine used, and there exists the effect of "loss of significance", well known in numerical mathematics and computational physics. The problem occurs primarily in numerical subtraction, and clearly, the estimation of derivatives involves the approximation of differences. In this article, we discuss several techniques for the estimation of derivatives. As a novel aspect, we divide into local and global methods, and explain the respective shortcomings. We have developed a general scheme for global methods, and illustrate our ideas by spline smoothing and spectral smoothing. The results from these less known techniques are confronted with the ones from local methods. As typical for the latter, we chose Savitzky-Golay-filtering and finite differences. Two basic quantities are used for characterization of results: The variance of the difference of the true derivative and its estimate, and as important new characteristic, the smoothness of the estimate. We apply the different techniques to numerically produced data and demonstrate the application to data from an aeroacoustic experiment. As a result, we find that global methods are generally preferable if a smooth process is considered. For rough estimates local methods work acceptably well.
We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization, characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength.PACS numbers: 05.45.-a, 63.50.-x, 63.70.+h The studies of ergodicity and dynamical thermalization in regular nonlinear lattices have a long history initiated by the Fermi-Pasta-Ulam problem [1] but they are still far from being complete (see, e.g., [2] for thermal transport in nonlinear chains and [3] for thermalization in a Bose-Hubbard model). In this letter, we study how the dynamical thermalization appears in nonlinear disordered chains where all linear modes are exponentially localized. Such modes appear due to the Anderson localization, introduced in the context of electron transport in disordered solids [4,5,6] and describing various physical situations like wave propagation in a random medium [7], localization of a Bose-Einstein condensate [8] and quantum chaos [9].Effects of nonlinearity on localization properties have attracted large interest recently. Indeed, nonlinearity naturally appears for localization of a Bose-Einstein condensate, as its evolution is described by the nonlinear Gross-Pitaevskii equation [10]. An interplay of disorder, localization, and nonlinearity is also important in other physical systems like wave propagation in nonlinear disordered media [11,12] and chains of nonlinear oscillators with randomly distributed frequencies [13].The main question here is whether the localization is destroyed by nonlinearity. It has been addressed recently using two physical setups. In refs. [14,15]it was demonstrated that an initially concentrated wavepacket spreads apparently indefinitely, although subdiffusively, in a disordered nonlinear lattice. For a transmission through a nonlinear disordered layer [16,17], chaotic destruction of localization leads to a drastically enhanced transparency.Here we study the thermalization properties of the dynamics of a nonlinear disordered lattice -discrete Anderson nonlinear Schrödinger equation (DANSE). We describe in details the features of the time-evolution of an initially localized excitation towards a statistical equilibrium in a finite lattice (we stress that this evolution is purely deterministic -and the relaxation to equilibrium is due to determinsitc chaos.). Below we argue that a statistically stationary state is characterized by the Gibbs energy equipartition across the linear eigenmodes (Eq. (5)) and call a relaxation to such an equilibrium state thermalization. Because thermalization is due to deterministic chaos, its rate heavily dependes on the statistical properties of the chaos. As is typical for nonlinear Hamiltonian systems, depending on initial conditions one can obtain solutions belonging to a "chaotic sea" or to "regul...
To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law.
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