Results 1  10
of
13
Pulse Propagation in Nonlinear Optical FiberLines That Employ PhaseSensitive Parametric Amplifiers
, 1994
"... Recently we have proposed using periodicallyspaced, phase sensitive optical parametric amplifiers to balance linear loss in a nonlinear fiberoptic communication line [Opt. Lett. 18, 803 (1993)]. Here we present a detailed analysis of pulse propagation in such a fiber line. Our analysis and numeric ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Recently we have proposed using periodicallyspaced, phase sensitive optical parametric amplifiers to balance linear loss in a nonlinear fiberoptic communication line [Opt. Lett. 18, 803 (1993)]. Here we present a detailed analysis of pulse propagation in such a fiber line. Our analysis and numerical simulations show that the length scale over which the pulse evolution occurs is significantly increased beyond a soliton period. This is because of the attenuation of phase variations across the pulse's profile by the amplifiers. Analytical evidence is presented which indicates that stable pulse evolution occurs on length scales much longer than the soliton period. This is confirmed through extensive numerical simulation, and the region of stable pulse propagation is found. The average evolution of such pulses is governed by a fourthorder nonlinear diffusion equation which describes the exponential decay of arbitrary initial pulses onto stable, steadystate, solitonlike pulses. Permane...
Dynamics of neural populations: Stability and synchrony
 Network: Comput. Neural Syst
, 2006
"... A population formulation of neuronal activity is employed to study an excitatory network of (spiking) neurons receiving external input as well as recurrent feedback. At relatively low levels of feedback, the network exhibits time stationary asynchronous behavior. A stability analysis of this time st ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
A population formulation of neuronal activity is employed to study an excitatory network of (spiking) neurons receiving external input as well as recurrent feedback. At relatively low levels of feedback, the network exhibits time stationary asynchronous behavior. A stability analysis of this time stationary state leads to an analytical criterion for the critical gain at which time asynchronous behavior becomes unstable. At instability the dynamics can undergo a supercritical Hopf bifurcation and the population passes to a synchronous state. Under different conditions it can pass to synchrony through a subcritical Hopf bifurcation. And at high gain a network can reach a runaway state, in finite time, after which the network no longer supports bounded solutions. The introduction of time delayed feedback leads to a rich range of phenomena. For example, for a given external input, increasing gain produces transition from asynchrony, to synchrony, to asynchrony and finally can lead to divergence. Time delay is also shown to strongly mollify the amplitude of synchronous oscillations. Perhaps, of general importance, is the result that synchronous behavior can exist only for a narrow range of time delays, which range is an order of magnitude smaller than periods
Stability Of Pulses In Nonlinear Optical Fibers Using PhaseSensitive Amplifiers
 SIAM J. Appl. Math
, 1996
"... . We consider the stability of solitonlike pulses propagating in nonlinear optical fibers with periodicallyspaced phasesensitive amplifiers, a situation where the averaged pulse evolution is governed by a fourthorder nonlinear diffusion equation similar to the KuramotoSivashinsky or SwiftHohenb ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
. We consider the stability of solitonlike pulses propagating in nonlinear optical fibers with periodicallyspaced phasesensitive amplifiers, a situation where the averaged pulse evolution is governed by a fourthorder nonlinear diffusion equation similar to the KuramotoSivashinsky or SwiftHohenberg equations. A bifurcation and stability analysis of this averagedequation is carried out, and in the limit of small amplifier spacing, a steadystate pulse solution is shown to be asymptotically stable. Furthermore, both a saddlenode bifurcation and a subcritical bifurcation from the zero solution are found. Analytical results are confirmed using the bifurcation software package auto. The analysis provides evidence for the existence of stable pulse solutions for a wide range of parameter values, including those corresponding to physically realizable soliton communications systems. Key words. solitons, nonlinear optical pulse propagation, optical fibers, bifurcation theory AMS subject c...
On the Asymptotic and Numerical Analysis of Exponentially IllConditioned Singularly Perturbed Boundary Value Problems
, 1995
"... Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions, and ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions, and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [SIAM J. Math. Anal. 11, (1980), pp. 122], is used to analytically calculate high order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [SIAM J. Appl. Math. 32, (1977), pp. 588597]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are show...
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation
 J. App. Math. Stoch. Anal., 2008. Article ID
"... This study considers the solution of a class of linear systems related with the fractional Poisson equation �FPE � �−∇2 � α/2 ϕ � g�x, y � with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to genera ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This study considers the solution of a class of linear systems related with the fractional Poisson equation �FPE � �−∇2 � α/2 ϕ � g�x, y � with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f�A � �A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f�A�b ≈ β0Vmf�Tm�e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thickrestart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions. Copyright q 2008 M. Ilić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Automatic Differentiation as a Tool for Sensitivity Analysis of a Convective Storm in a 3D Cloud Model
"... The ADIFOR automatic differentiation tool is applied to a 3D stormscale meteorological model to generate a sensitivityenhanced code capable of providing derivatives of all model output variables and related diagnostic (derived) parameters as a function of specified control parameters. The tangent ..."
Abstract
 Add to MetaCart
The ADIFOR automatic differentiation tool is applied to a 3D stormscale meteorological model to generate a sensitivityenhanced code capable of providing derivatives of all model output variables and related diagnostic (derived) parameters as a function of specified control parameters. The tangent linear approximation, applied to a deep convective storm by the first of its kind using a fullphysics compressible model, is valid up to 50 min for a 1 % water vapor perturbations. The result is very encouraging considering the highly nonlinear and discontinuous properties of solutions. The ADIFORgenerated code has provided valuable sensitivity information on storm dynamics. Especially, it is very efficient and useful for investigating how a perturbation inserted at earlier time propagates through the model variables at later times. However, it is computationally very expensive to be applied to the variational data assimilation, especially for 3D meteorological models, which potentially ...
A Functional Analytic Approach to Transient Signal Detection and Estimation
, 1989
"... This dissertation takes a functionalanalytic approach to the problem of transient signal detection and estimation. Here, the class of transient signals is viewed as a nonparametric subspace of an appropriate Hilbert space. Specifically, we consider the generalized class of transient signals in a So ..."
Abstract
 Add to MetaCart
This dissertation takes a functionalanalytic approach to the problem of transient signal detection and estimation. Here, the class of transient signals is viewed as a nonparametric subspace of an appropriate Hilbert space. Specifically, we consider the generalized class of transient signals in a Sobolev space setting which consists of timelimited and bandlimited signals. In this setting, we show that the optimal estimator, in the "mixednorm" sense, for the general class of transient signals both in time and frequency domains is a subclass of Lspline functions. Using this fact together with a correspondence between spline smoothing problems and that of leastsquare estimation problems, the transient signal estimation problem is formulated in a statespace approach, thereby allowing us to perform transient signal estimation procedures in a recursive manner. To evaluate the performance of these types of nonparametric estimators, we consider several transient signal models contaminated by white noise. We use a spline smoother as the optimal estimator for the modelled class of transient signals. Results from a Monte Carlo study demonstrate that the estimation procedure possesses an excellent capability for reconstructing the signal. (See Figure 1.1 and related comments below.) We then use these spline smoothing estimates to derive a threshold detector for transient signals...
unknown title
, 2003
"... Surfaces roughness effects on the transmission of Gaussian beams by anisotropic parallel plates ..."
Abstract
 Add to MetaCart
Surfaces roughness effects on the transmission of Gaussian beams by anisotropic parallel plates