2,793 research outputs found
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling
We analyze the effect of adding a weak, localized, inhomogeneity to a two
dimensional array of oscillators with nonlocal coupling. We propose and also
justify a model for the phase dynamics in this system. Our model is a
generalization of a viscous eikonal equation that is known to describe the
phase modulation of traveling waves in reaction-diffusion systems. We show the
existence of a branch of target pattern solutions that bifurcates from the
spatially homogeneous state when , the strength of the
inhomogeneity, is nonzero and we also show that these target patterns have an
asymptotic wavenumber that is small beyond all orders in .
The strategy of our proof is to pose a good ansatz for an approximate form of
the solution and use the implicit function theorem to prove the existence of a
solution in its vicinity. The analysis presents two challenges. First, the
linearization about the homogeneous state is a convolution operator of
diffusive type and hence not invertible on the usual Sobolev spaces. Second, a
regular perturbation expansion in does not provide a good ansatz
for applying the implicit function theorem since the nonlinearities play a
major role in determining the relevant approximation, which also needs to be
"correct" to all orders in . We overcome these two points by
proving Fredholm properties for the linearization in appropriate Kondratiev
spaces and using a refined ansatz for the approximate solution, which obtained
using matched asymptotics.Comment: 39 pages, 1 figur
On the spectra of certain integro-differential-delay problems with applications in neurodynamics
We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs
Hydrodynamic dispersion within porous biofilms
Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport
Correspondence between one- and two-equation models for solute\ud transport in two-region heterogeneous porous media
In this work, we study the transient behavior of upscaled models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the time scale of interest (t) is smaller than the characteristic time (T1) for the relaxation of the effective macroscale parameters (i.e., when t †T1); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (T1) (i.e., when t » T1); and (3) a one-equation, time-asymptotic formulation (1eqâ). This model can be adopted when (t) is significantly larger than the time scale (T2) associated with exchange processes between the two regions (i.e., when t » T2). In order to obtain some physical insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in simple cases. The main result of this paper is to show that there is weak long-time convergence of the solution of (2eq) toward the solution of (1eqâ) in terms of standardized moments but, interestingly, not in terms of centered moments. Physically, our interpretation of this result is that the spreading of the solute is dominating higher order non-zero perturbations in the asymptotic regime
Patchiness and Demographic Noise in Three Ecological Examples
Understanding the causes and effects of spatial aggregation is one of the
most fundamental problems in ecology. Aggregation is an emergent phenomenon
arising from the interactions between the individuals of the population, able
to sense only -at most- local densities of their cohorts. Thus, taking into
account the individual-level interactions and fluctuations is essential to
reach a correct description of the population. Classic deterministic equations
are suitable to describe some aspects of the population, but leave out features
related to the stochasticity inherent to the discreteness of the individuals.
Stochastic equations for the population do account for these
fluctuation-generated effects by means of demographic noise terms but, owing to
their complexity, they can be difficult (or, at times, impossible) to deal
with. Even when they can be written in a simple form, they are still difficult
to numerically integrate due to the presence of the "square-root" intrinsic
noise. In this paper, we discuss a simple way to add the effect of demographic
stochasticity to three classic, deterministic ecological examples where
aggregation plays an important role. We study the resulting equations using a
recently-introduced integration scheme especially devised to integrate
numerically stochastic equations with demographic noise. Aimed at scrutinizing
the ability of these stochastic examples to show aggregation, we find that the
three systems not only show patchy configurations, but also undergo a phase
transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy
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