random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

the global attractor of g navier stokes equations with linear dampness on r2 in this paper the long time behaviors of g navier stokes equations with linear dampness on r 2 were investigated by using the energy equation method the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted sobolev space moreover the estimation of the hausdorff and fractal dimensions of such attractors were also obtained

d pullback attractor for a non autonomous wave equation with additive noise on unbounded domains this paper is devoted to investigating the asymptotic behavior of solutions of the wave equation with additive noise as well as a non autonomous term defined on unbounded domains firstly by uniform estimates a closed d pullback absorbing set of dynamical system f produced by the mentioned wave equation is obtained secondly the asymptotical compactness of f is established by the splitting technique finally the existence of ad pullback attractor for corresponding dynamical system f in h 1 r 3 xl 2 r 3 is proved where n 3 particularly we can show that the d pullback attractor is also periodic when the forcing term is periodic

pullback attractors for the non autonomous fitzhugh nagumo system on unbounded domains abstract the existence of a pullback attractor is established for the singularly perturbed fitzhugh nagumo system defined on the entire space r n when external terms are unbounded in a phase space the pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far field values of solutions although the limiting system has no global attractor we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded and hence do not blow up as a small parameter approaches zero

on the backward behavior of some dissipative evolution equations abstract we prove that every solution of a kdv burgers sivashinsky type equation blows up in the energy space backward in time provided the solution does not belong to the global attractor this is a phenomenon contrast to the backward behavior of the periodic 2d navier stokes equations studied by constantin et al 1997 but analogous to the backward behavior of the kuramoto sivashinsky equation discovered by kukavica and malcok 2005 also we study the backward behavior of solutions to the damped driven nonlinear schrodinger equation the complex ginzburg landau equation and the hyperviscous navier stokes equations in addition we provide some physical interpretation of various backward behaviors of several perturbations of the kdv equation by studying explicit cnoidal wave solutions furthermore we discuss the connection between the backward behavior and the energy spectra of the solutions the study of backward behavior of dissipative evolution equations is motivated by the investigation of the bardos tartar conjecture on the navier stokes equations stated in bardos and tartar 1973

global attractor for weakly damped nonlinear schrodinger equations in l2 r abstract we prove that the weakly damped nonlinear schrodinger flow in l 2 r provides a dynamical system which possesses a global attractor the proof relies on the continuity of the schrodinger flow for the weak topology in l 2 r

asymptotic smoothing and attractors for the generalized benjamin bona mahony equation on r 3 abstract we study the asymptotic behavior of the solutions of the benjamin bona mahony equation defined on r 3 we first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain which involves the littlewood paley projection operators we then prove the existence of an attractor for the benjamin bona mahony equation in the phase space h 1 r 3 by showing the solutions are point dissipative and asymptotic compact finally we establish the regularity of the attractor and show that the attractor is bounded in h 2 r 3

long time behavior for the weakly damped driven long wave short wave resonance equations abstract in this paper we consider the cauchy problem of the long wave short wave resonance equations by making use of a strichartz type inequality for the solutions decomposing suitably the solution semigroup into a decay parts and a more regular parts and ruling out the vanishing and dichotomy of the solutions we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions

asymptotic smoothing effect for a weakly damped nonlinear schrodinger equation in t2 abstract we prove that the global attractor for a weakly damped two dimensional nonlinear schrodinger equation in the usual energy space is in fact included and compact in a more regular energy space the method relies on a suitable approximation when time goes to infinity of the high frequency modes of the solutions

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

monotonicity and boundedness in implicit runge kutta methods this paper concerns the analysis of implicit runge kutta methods for approximating the solutions to stiff initial value problems the analysis includes the case of nonlinear systems of differential equations that are essentially more general than the classical test equationu u with a complex constant the properties of monotonicity and boundedness of a method refer to specific moderate rates of growth of the approximations during the numerical calculations this paper provides necessary conditions for these properties by using the important concept of algebraic stability introduced by burrage butcher and by crouzeix these properties will also be related to the concept of contractivity b stability and to a weakened version of contractivity

long time behavior of solution for coupled ginzburg landau equations describing bose einstein condensates and nonlinear optical waveguides and cavities abstract in this paper the cauchy problem of coupled ginzburg landau gl equations describing bose einstein condensates and nonlinear optical waveguides and cavities is considered the long time behavior of solution is investigated using a series of sharp a priori estimates in phase space e 1 and e 2 respectively the global strong attractor in e 2 is proved by energy equation method and its bound of dimension is shown

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

the uniform attractor for the 2d non autonomous navier stokes flow in some unbounded domain abstract by using the energy equation method the existence of the uniform attractor for the 2d non autonomous navier stokes equations in some unbounded domains with usual settings is obtained without the restriction of the forcing term belonging to some weighted sobolev space and for the quasi periodic forcing term the estimation of the hausdorff dimension of such attractors is also obtained

extended capillary wave model for the liquid vapor interface and its width in the limit of vanishing gravity an extended capillary wave model is derived from density functional theory and the predictions of the usual capillary wave model are reexamined the model is expressed in terms of the direct correlation function c r r of the equilibrium fluid a small wavevector expansion of c r r reduces the extended model to the usual one but it is shown that higher order terms also play a role in determining interfacial properties in the limit of vanishing gravity the meaning of the root mean square fluctuation w of the distortions of the gibbs dividing surface and its relation to the actual width of the density profile are specified for spatial dimensions d 3 the predictions of the extended and the usual models agree explicit calculations for d 3 show w in lc a with lc the classical capillary length and a a length proportional to the fourth transverse moment of c r r adoption of a scaling hypothesis near g 0 similar to that proposed by weeks shows that a lc and w does not diverge as g 0 it is argued that this does not necessarily imply that the width is also finite although this remains a possibility extensions to week s hypothesis are also examined

regularity of the global attractor and finite dimensional behavior for the second grade fluid equations abstract this paper is devoted to the large time behavior and especially to the regularity of the global attractor of the second grade fluid equations in the two dimensional torus we first recall that for any size of the material coefficient 0 these equations are globally well posed and admit a compact global attractor a in h 3 t 2 2 we prove that for any 0 there exists 0 such that a belongs to h 3 t 2 2 if the forcing term is in h 1 t 2 2 we also show that this attractor is contained in any sobolev space h 3 m t 2 2 provided that is small enough and the forcing term is regular enough these arguments lead also to a new proof of the existence of the compact global attractor a furthermore we prove that on a the second grade fluid system can be reduced to a finite dimensional system of ordinary differential equations with an infinite delay moreover the existence of a finite number of determining modes for the equations of the second grade fluid is established

global attractors for the klein gordon schrodinger equation in unbounded domains abstract in this paper we study the long time behavior of solutions for the klein gordon schrodinger equation in the whole space r n with n 3 we first prove the continuity of the solutions on initial data and then establish the asymptotic compactness of solutions finally we show the existence of the global attractor for this model in the space hk r n hk r n for each integer k 1

dynamics of stochastic zakharov equations the asymptotic behavior of zakharov equations driven by random force is studied the force which is smooth enough and homogeneous in space and white noise in time acts on both equations of the zakharov system independently by some a priori estimates we prove the existence of a solution in energy spaces e1 and e2 via the galerkin approximation this solution is defined on the given probability space rather than a martingale solution then a global random attractor is constructed in energy space e2 equipped with weak topology further the existence of a stationary measure is proved in energy space e2 with usual topology

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

stability properties for a general class of methods for ordinary differential equations the concepts of g stability for linear multistep methods and b stability for runge kutta methods are combined in a unified approach to nonlinear stability for a class of methods general enough to include these as special cases recent work on these developments by kevin burrage and the author is reviewed and to exemplify the new ideas and techniques an algebraic stability analysis is presented for the backward differentiation methods of orders 1 2 and 3

on the nonlinear schrodinger limit of the magnetic zakharov system abstract in this paper we study the limit behavior of a smooth solution for the magnetic type zakharov system as the parameters tend to infinity we prove that solutions of the magnetic zakharov system converge to the solution of the nonlinear schrodinger equation moreover the detailed convergence rate is also investigated

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

finite dimensional global attractor for a dissipative anisotropic fourth order schrodinger equation abstract we study the long time behavior of the solutions to a nonlinear damped anisotropic fourth order schrodinger type equation in r 2 we prove that this behavior is described by the existence of regular finite dimensional global attractor in the energy space

regularity of attractors for the benjamin bona mahony equation this paper deals with the regularity of the global attractor for the benjamin bona mahony equation we prove that the global attractor is smooth if the forcing term is smooth

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

finite dimensional global attractor for dissipative schrodinger boussinesq equations abstract in this paper the authors consider the initial boundary value problems of dissipative schrodinger boussinesq equations and prove the existence of global attractors and the finiteness of the hausdorff and the fractal dimensions of the attractors

crossover from extensive to nonextensive behavior driven by long range d 1 bond percolation abstract we present a monte carlo study of a linear chain d 1 with long range bonds whose occupancy probabilities are given by p ij p r ij 0 p 1 0 where rij 1 2 is the distance between sites the 0 corresponds to the first neighbor mean field particular case we exhibit that the order parameter p equals unity p 0 for 0 1 presents a familiar behavior i e 0 for p pc and finite otherwise for 1 2 our results confirm recent conjecture namely that the nonextensive region 0 1 can be meaningfully unfolded as well as unified with the extensive region 1 by exhibiting p as a function of p where 1 p 1 p n n n 1 d 1 1 d n being the number of sites of the chain a corollary of this conjecture now numerically verified is that pc 1 in the 1 0 limit

efficient and stable numerical methods for the generalized and vector zakharov system we present efficient and stable numerical methods for approximations of the generalized zakharov system gzs and vector zakharov system for multicomponent plasma vzsm with without a linear damping term the key points in the methods are based on i a time splitting discretization of a schrodinger type equation in gzs or vzsm ii discretizing a nonlinear wave type equation by a pseudospectral method for spatial derivatives and iii solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the methods are explicit unconditionally stable and of spectral order accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping term in gzs or vzsm conserve or keep the same decay rate of the wave energy as that in gzs or vzsm without a or with a linear damping term and give exact results for the plane wave solution extensive numerical tests are presented for plane waves and solitary wave collisions in one dimensional gzs and we also give the dynamics of three dimensional vzsm to demonstrate our new efficient and accurate numerical methods furthermore the methods are applied to study the convergence and quadratic convergence rates of vzsm to gzs and of gzs to the nonlinear schrodinger nls equation in the subsonic limit regime 0 varepsilon ll1 where the parameter varepsilon is inversely proportional to the acoustic speed our tests also suggest that the following meshing strategy or varepsilon resolution is admissible in this regime spatial mesh size h o varepsilon and time step k o varepsilon

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

low dimensional chaos in a driven damped nonlinear schro dinger equation abstract we study bifurcations of attractors in a driven damped nonlinear schrodinger equation which models nonlinear responses of a plasma driven by a rf field and a sequence of period doubling bifurcations is shown to lead to a chaotic attractor in a low dimensional subspace spanned by a soliton and long wavelength radiation

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

analogs of renormalization group transformations in random processes abstract we review the properties of a real space renormalization group transformation of the free energy including the existence of oscillatory terms multiplying the non analytic part of the free energy we then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation the essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary these random processes are shown to be generated directly from cantor sets in a spatial representation the ensuing random process exhibits a transition between gaussian and fractal behavior in the fractal regime the trajectories will in an average sense form self similar clusters in a temporal representation the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior the fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

stability and b convergence of general linear methods abstract this paper continues earlier work by the same author concerning the stability and b convergence of general linear methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space in a previous paper we have proved that bh consistency resp bh consistency together with bh stability implies optimal b convergence resp b convergence in this paper by means of bs and bsi stability properties the sufficient conditions for a method to be b bh bh or bh consistent bh or weakly bh stable b or optimally b convergent respectively are further established

a determining form for the damped driven nonlinear schrodinger equation fourier modes case in this paper we show that the global attractor of the 1d damped driven nonlinear schrodinger equation nls is embedded in the long time dynamics of a determining form the determining form is an ordinary differential equation in a space of trajectories x cb1 r pmh2 where pm is the l2 projector onto the span of the first m fourier modes there is a one to one identification with the trajectories in the global attractor of the nls and the steady states of the determining form we also give an improved estimate for the number of the determining modes

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

asymptotic smoothing effect of solutions to weakly dissipative klein gordon schrodinger equations abstract in this paper the authors consider the cauchy problem of weakly dissipative klein gordon schrodinger equations through yukawa coupling in r 3 making use of a strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions

the renormalization group and optimization of entropy we illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group rg approach when applied to systems with scaling symmetry we consider three examples 1 gaussian fixed point criticality in a fluid or in the capillary wave model of an interface 2 levy like random walks with self similar cluster formation and 3 long ranged bond percolation in all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point we use an equivalence between random walk distributions and order parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long tailed distributions

random attractors for the stochastic fitzhugh nagumo system on unbounded domains abstract the existence of a random attractor for the stochastic fitzhugh nagumo system defined on an unbounded domain is established the pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables these estimates are obtained by a cut off technique

pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on two dimensional unbounded domains this paper studies the pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on r l l for some l 0 we first use the technique of truncation functions together with the decomposition of spatial domain to prove the existence of a pullback attractor in then we discuss the upper semicontinuity of the pullback attractors when the spatial domains vary from m to where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m 1 m let a a r and am am r be the pullback attractors corresponding to and m respectively we establish for each r that for any neighborhood o a of a am enters o a if m is large enough

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

long time behavior of the solutions for the dissipative modified zakharov equations for plasmas with a quantum correction abstract in this paper we study the long time behavior of the solutions for the dissipative modified zakharov equations for plasmas with a quantum correction subsequent to our previous work l y jin et al 2012 where the existence of the global attractor a l 2 h 2 h 0 1 h 3 h 0 1 is obtained via delicate a prior estimates we are further going to investigate and obtain the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor