stability of finite difference schemes for complex diffusion processes in this paper we present a rigorous proof for the stability of a class of finite difference schemes applied to nonlinear complex diffusion equations complex diffusion is a common and broadly used denoising procedure in image processing to illustrate the theoretical results we present some numerical examples based on an explicit scheme applied to a nonlinear equation in the context of image denoising a correction is attached

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

a fourth order explicit schemes for the coupled nonlinear schrodinger equation abstract in this paper we derive numerical methods for solving the coupled nonlinear schrodinger equation we discretize the space derivative by central difference formulas of fourth order the resulting ordinary differential system is solved by the fourth order explicit runge kutta method neumann and periodic boundary conditions are used the method is tested for accuracy and the conserved quantities these methods conserve the three conserved quantities conserve exactly for at least five decimal places a comparison has been made with some existing methods

stability analysis of difference schemes for variable coefficient schro dinger type equations we consider the stability of difference schemes for the solution of the initial boundary value problem for the equation u 1 left a x t u x right x b x t u c x t u f x t where u a b c and f are complex valued functions using energy methods we establish the stability of a general two level scheme which includes euler s method crank nicolson s method and the backward euler method if the coefficient a x t is purely imaginary the explicit euler method is unconditionally unstable for this case we propose a new scheme with appropriately chosen artificial dissipation which we prove to be conditionally stable

lorenzian analysis of infinite poissonian populations and the phenomena of paretian ubiquity abstract the lorenz curve is a universally calibrated statistical tool measuring quantitatively the distribution of wealth within human populations we consider infinite random populations modeled by inhomogeneous poisson processes defined on the positive half line the randomly scattered process points representing the wealth of the population members or any other positive valued measure of interest such as size mass energy etc for these populations the notion of macroscopic lorenz curve is defined and analyzed and the notion of lorenzian fractality is defined and characterized we show that the only non degenerate macroscopically observable lorenz curves are power laws manifesting paretian statistics thus providing a universal lorenzian explanation to the ubiquitous appearance of paretian probability laws in nature

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

fractal poisson processes abstract the central limit theorem clt and extreme value theory evt study respectively the stochastic limit laws of sums and maxima of sequences of independent and identically distributed i i d random variables via an affine scaling scheme in this research we study the stochastic limit laws of populations of i i d random variables via nonlinear scaling schemes the stochastic population limits obtained are fractal poisson processes which are statistically self similar with respect to the scaling scheme applied and which are characterized by two elemental structures i a universal power law structure common to all limits and independent of the scaling scheme applied ii a specific structure contingent on the scaling scheme applied the sum projection and the maximum projection of the population limits obtained are generalizations of the classic clt and evt results extending them from affine to general nonlinear scaling schemes

a galerkin method for a nonlinear integro differential wave system a galerkin method is introduced for the numerical solution of a system equivalent to the equation tt 1 0 x2dx xx which models the vibrations of a string when the increase in tension due to extension is taken into account we prove the convergence of the semidiscrete approximations and present some numerical tests

universal self similarity of propagating populations this paper explores the universal self similarity of propagating populations the following general propagation model is considered particles are randomly emitted from the origin of a d dimensional euclidean space and propagate randomly and independently of each other in space all particles share a statistically common yet arbitrary motion pattern each particle has its own random propagation parameters emission epoch motion frequency and motion amplitude the universally self similar statistics of the particles displacements and first passage times fpts are analyzed statistics which are invariant with respect to the details of the displacement and fpt measurements and with respect to the particles underlying motion pattern analysis concludes that the universally self similar statistics are governed by poisson processes with power law intensities and by the fr echet and weibull extreme value laws

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

fractal probability laws we explore six classes of fractal probability laws defined on the positive half line weibull frechet levy hyper pareto hyper beta and hyper shot noise each of these classes admits a unique statistical power law structure and is uniquely associated with a certain operation of renormalization all six classes turn out to be one dimensional projections of underlying poisson processes which in turn are the unique fixed points of poissonian renormalizations the first three classes correspond to linear poissonian renormalizations and are intimately related to extreme value theory weibull frechet and to the central limit theorem levy the other three classes correspond to nonlinear poissonian renormalizations pareto s law commonly perceived as the universal fractal probability distribution is merely a special case of the hyper pareto class

a tau method based on non uniform space time elements for the numerical simulation of solitons abstract in this paper we discuss the numerical simulation of soliton solutions of schrodnger s system of nonlinear partial differential equations the technique used is based on a novel approach to the tau method based on space time elements the original operator is not discretized in either space or time the perturbation term which in the tau method is a by product of the computation is used to measure the error in the equation or defect and therefore to control the accuracy of the computation in an adaptive way this leads to an accurate and economical numerical procedure which even for approximations of a very low degree can simulate surfaces with very sharp gradients in the direction of both space and time the tau method formulation presented in this paper preserves very accurately the principle of conservation of energy however our numerical examples support the idea that the latter is a necessary but not a sufficient condition to guarantee an efficient approximation of soliton solutions of schrodinger s equation by means of a method of numerical simulation

dufort frankel type methods for linear and nonlinear schrodinger equations in this paper we propose a so called dufort frankel type method for the linear and nonlinear schrodinger equations schemes of the dufort frankel type are explicit and have solution independent stability criteria moreover the dufort frankel type method satisfies a discrete conservation law for the schrodinger equations and this is important to get reliable numerical solutions

scale invariance of random populations from paretian to poissonian fractality random populations represented by stochastically scattered collections of real valued points are abundant across many fields of science fractality in the context of random populations is conventionally associated with a paretian distribution of the population s values using a poissonian approach to the modeling of random populations we introduce a definition of poissonian fractality based on the notion of scale invariance this definition leads to the characterization of four different classes of fractal poissonian populations three of which being non paretian objects the fractal poissonian populations characterized turn out to be the unique fixed points of natural renormalizations and turn out to be intimately related

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

a unified and universal explanation for levy laws and 1 f noises levy laws and 1 f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals the signals are considered to follow statistically a common yet arbitrary generic signal pattern which may be either stationary or dissipative each signal is considered to have its own random transmission amplitude and frequency we characterize the amplitude frequency randomizations which render the system output s stationary law and power spectrum universal i e independent of the underlying generic signal pattern the classes of universal stationary laws and power spectra are shown to coincide respectively with the classes of levy laws and 1 f noises thus providing a unified and universal explanation for the ubiquity of these classes of anomalous statistics in various fields of science and engineering

simple time and space adaptation in one dimensional evolutionary partial differential equations a simple technique for time and space adaptation in one dimensional evolutionary partial differential equations suggested by sanz serna and christie 1 is tested it is found that the equidistribution of hu xi greatly improves on the equidistribution of arc length used by those authors the time step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed

a compact finite difference scheme for the nonlinear schrodinger equation with wave operator abstract in this paper a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear schrodinger nls equation with wave operator this is a scheme of three levels with a discrete conservation law the unconditional stability and convergence in maximum norm with order o h 4 2 are proved by the energy method a numerical experiment is presented to support our theoretical results

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

on the development of a coupled nonlinear telegraph diffusion model for image restoration abstract in this work we propose a telegraph coupled partial differential equation tcpde based model for image restoration the new framework interpolates between a couple of nonlinear telegraph equation and a parabolic equation this proposed strategy can be applied to significantly preserve the oscillatory and texture pattern in an image even in low signal to noise ratio snr first we prove that the present model has a unique global weak solution using banach s fixed point theorem then we apply our model over a set of gray level images to illustrate the image denoising ability of the proposed model the experimental results of the proposed model are reported which found better in terms of noise elimination and edge preservation with respect to the recently developed hyperbolic parabolic pde based models as well as coupled diffusion based model

solving the generalized nonlinear schrodinger equation via quartic spline approximation this paper is concerned with a new conservative finite difference method for solving the generalized nonlinear schrodinger gnls equation iut uxx f u 2 u 0 the numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation central difference and extrapolation formulae are used for approximating the neumann boundary conditions introduced both continuous and discrete energy conservation and the stability property are investigated the numerical method provides an efficient and reliable way for computing long time solitary solutions given by the gnls equation numerical examples are given to demonstrate our conclusions

a conservative numerical scheme for a class of nonlinear schrodinger equation with wave operator in this paper the initial boundary value problem of a class of nonlinear schrodinger equation with wave operator is considered an explicit and efficient finite difference scheme is presented this is a scheme of four levels with a discrete conservative law convergence and stability are proved

analysis of a leap frog pseudospectral scheme for the schrodinger equation the numerical properties of a leap frog pseudospectral scheme for the schrodinger equation are analyzed stability second order accuracy in time and spectral accuracy in space are discussed considering the linear schrodinger equation with potential in a periodic setting further issues regarding phase error gauge invariance conservation properties and commutation relations are addressed results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known crank nicholson finite difference scheme

an adaptive mesh refinement method for nonlinear dispersive wave equations abstract adaptive mesh refinement techniques are often essential for solving nonlinear partial differential equations numerically a new method for spatial grid refinement is developed and implemented several numerical experiments are performed to compare the method with results obtained using a uniform grid the new method has the following properties it is simple to implement and requires little modification of existing code to use the solutions achieved as a result of using these methods prove to be accurate and the stability of the numerical methods is affected minimally the effect of the grid refinement on essential properties of some of the equations such as conservation is minimized through the use of piecewise uniformity

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

on the structure and phase transitions of power law poissonian ensembles power law poissonian ensembles are poisson processes that are defined on the positive half line and that are governed by power law intensities power law poissonian ensembles are stochastic objects of fundamental significance they uniquely display an array of fractal features and they uniquely generate a span of important applications in this paper we apply three different methods oligarchic analysis lorenzian analysis and heterogeneity analysis to explore power law poissonian ensembles the amalgamation of these analyses combined with the topology of power law poissonian ensembles establishes a detailed and multi faceted picture of the statistical structure and the statistical phase transitions of these elemental ensembles

diversity of poissonian populations populations represented by collections of points scattered randomly on the real line are ubiquitous in science and engineering the statistical modeling of such populations leads naturally to poissonian populations poisson processes on the real line with a distinguished maximal point poissonian populations are infinite objects underlying key issues in statistical physics probability theory and random fractals due to their infiniteness measuring the diversity of poissonian populations depends on the lower bound cut off applied this research characterizes the classes of poissonian populations whose diversities are invariant with respect to the cut off level applied and establishes an elemental connection between these classes and extreme value theory the measures of diversity considered are variance and dispersion simpson s index and inverse participation ratio shannon s entropy and renyi s entropy and gini s index

correspondence principle for the quantized annulus romanovski polynomials and morse potential abstract a general theory of quantization has been proposed by berezin see e g his survey in comm math phys 40 1975 153 174 in this paper we establish a weak form of the correspondence principle for the annulus quantized according to berezin more precisely we show that b i as 0 where i is the identity operator and b the berezin transform we consider also spectral analysis on the annulus in particular we express the eigenfunctions of the relevant laplacian in terms of certain romanovski polynomials finally we write down the expression for the analogue of the morse potential in this case we remark that similar considerations can be made in principle on any circular domain in the presence of a radial hermitian metric of constant curvature

an unconditionally stable three level explicit difference scheme for the schro dinger equation with a variable coefficient a three level explicit difference scheme for the schrodinger equation with a variable coefficient is established it is proved by the discrete energy method that the scheme is unconditionally stable

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

parametric spline schemes for the coupled nonlinear schrodinger equation abstract in this study the parametric cubic spline scheme is implemented to find the approximate solution of the coupled nonlinear schrodinger equations this scheme is based on the crank nicolson method in time and parametric cubic spline functions in space the error analysis and stability of the scheme are investigated and the numerical results show that we can get different precision schemes by choosing suitably parameter values and this scheme is unconditionally stable two problems are solved to illustrate the efficiency of the methods as well as to compare with other methods

randomized central limit theorems a unified theory the central limit theorems clts characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables the clts assert that the universal probability laws governing ensembles aggregate statistics are either gaussian or levy and that the universal probability laws governing ensembles extreme statistics are frechet weibull or gumbel the scaling schemes underlying the clts are deterministic scaling all ensemble components by a common deterministic scale however there are random environment settings in which the underlying scaling schemes are stochastic scaling the ensemble components by different random scales examples of such settings include holtsmark s law for gravitational fields and the stretched exponential law for relaxation times in this paper we establish a unified theory of randomized central limit theorems rclts in which the deterministic clt scaling schemes are replaced with stochastic scaling schemes and present randomized counterparts to the classic clts the rclt scaling schemes are shown to be governed by poisson processes with power law statistics and the rclts are shown to universally yield the levy frechet and weibull probability laws

paretian poisson processes many random populations can be modeled as a countable set of points scattered randomly on the positive half line the points may represent magnitudes of earthquakes and tornados masses of stars market values of public companies etc in this article we explore a specific class of random such populations we coin paretian poisson processes this class is elemental in statistical physics connecting together in a deep and fundamental way diverse issues including the poisson distribution of the law of small numbers paretian tail statistics the frechet distribution of extreme value theory the one sided levy distribution of the central limit theorem scale invariance renormalization and fractality resilience to random perturbations

general recurrence and ladder relations of hypergeometric type functions a method for the explicit construction of general linear sum rules involving hypergeometric type functions and their derivatives of any order is developed this method only requires the knowledge of the coefficients of the differential equation that they satisfy namely the hypergeometric type differential equation special attention is paid to the differential recurrence or ladder relations and to the fundamental three term recurrence formulas most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials are instances of these sum rules moreover an extension of the method to the generalized hypergeometric type functions is also described allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen like atoms where the varying parameter is the dimensionality