pade numerical schemes for the sine gordon equation abstract the sine gordon equation turn up in several problems in science and engineering although it is integrable in practical applications its numerical solution is powerful and versatile four novel implicit finite difference methods based on q s pade approximations with q s th order in space have been developed and analyzed for this equation all share the same treatment for the nonlinearity and integration in time concretely 0 4 2 2 2 4 and 4 4 pade methods additionally the energy conserving strauss vazquez scheme has been considered in a 0 2 pade implementation these methods have been compared among them for both the kink antikink and breather solutions in terms of global error computational cost and energy conservation the 0 4 and 2 4 pade methods are the most cost effective ones for small and large global error respectively our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

testing misspecification in vector time series models with exogenous variables summary in this paper we consider the application of hypothesis testing procedures to armax systems the score or lagrangian multiplier principle is used to obtain misspecification tests appropriate to these models the structure of the information or sensitivity matrix is investigated in order to discuss the identifiability of alternative models and is used to show the existence of a set of admissible alternatives that are locally equivalent to one another this notion is then developed to suggest why the score test is to be regarded as a pure significance test

spectral decomposition of bell operators for multiqubit systems the spectral decomposition of multipartite bell operators for two dichotomic observables per site as introduced by werner and wolf phys rev a 64 032112 2001 is done implications on the characterization of bell inequalities as criteria of entanglement are discussed

vector autoregressive moving average identification for macroeconomic modeling a new methodology this paper develops a new methodology for identifying the structure of varma time series models the analysis proceeds by examining the echelon form and presents a fully automatic strongly consistent data driven approach to model specification a novel feature of the inferential procedures developed here is that they work in terms of a canonical representation based on the kronecker invariants in which the variables are expressed in the form of scalar dynamic structural equations derived from the varma system this feature facilitates the construction of procedures which from the perspective of macroeconomic modeling can be efficacious in that they do not rely on var approximations techniques that are applicable to both asymptotically stationary and unit root partially nonstationary cointegrated time series models are presented the inferential potential of the techniques is illustrated via simulation experiments that use data generating mechanisms based on real world examples drawn from the time series literature aspects of the kronecker invariants that impinge on practical application and that have not hitherto been discussed in the literature are explored

an approach to testing linear times series models the purpose of this paper is to develop diagnostic tests for open loop transfer function models with autoregressive moving average stochastic disturbances using the efficient scores procedure the tests proposed are asymptotically equivalent to those based upon the likelihood ratio principle and have the advantage that they do not involve a heavy computational burden consideration is given both to the standard case of nonsingular information matrices and to the situation obtaining when there are identifiability problems and application of conventional large sample test procedures is not feasible relationships between score tests and portmanteau tests in time series analysis are also investigated some simulation evidence on the finite sample behaviour of the tests is presented and it is seen that the tests of this paper perform well

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

frequency and phase locking of spatially periodic perturbed sine gordon breather trains this paper considers the damped driven sine gordon equation with even periodic boundary conditions on a finite length nearly integrable perturbation methods are applied to infer approximate solution branches of space and time periodic breather trains of order one amplitude these approximate solutions consist of coherent spatial structures either homogeneous or with one breather per period which lock in frequency and phase to the small time periodic perturbation in the presence of uniform weak damping the predicted resonant breather trains are then verified by overman via direct simulation of the perturbed sine gordon pde and shown to be accurate within the order of the perturbation theory

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

coherence and chaos in the driven damped sine gordon equation measurement of the soliton spectrum abstract a numerical procedure is developed which measures the sine gordon soliton and radiation content of any field t which is periodic in space the procedure is applied to the field generated by a damped driven sine gordon equation this field can be either temporally periodic locked to the driver or chaotic in either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized soliton wave train modes the numerical procedure quantitatively identifies the presence number and properties of these soliton wave trains for example an increase of spatial symmetry is accompanied by the injection of additional solitons into the field

the solitons of zabusky and kruskal revisited perspective in terms of the periodic spectral transform abstract we consider the numerical soliton experiments of zabusky and kruskal 1 and address their results in light of the spectral structure of the periodic korteweg devries kdv equation we derive and exploit a simple relationship between the infinite line transfer matrix and the monodromy matrix of periodic theory this establishes the connection at all stages of our analysis between the infinite line soliton and the soliton limit of periodic theory we develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic fourier analysis finally we introduce the concept of a reference level which is defined to be the level upon which the solitons propagate these results together with judicious numerical experiments allow us to reassess the work of zabusky and kruskal from the standpoint of periodic theory for the kdv equation we find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper we also find that even in the presence of periodic boundary conditions the solitons behave like infinite line solitons in that they undergo phase shifts upon collision with each other our estimate of the fpu recurrence time which includes the effect of phase shifting is within one percent of their observed value

rank based autoregressive order identification abstract optimal rank based procedures have been derived for testing arbitrary linear restrictions on the parameters of autoregressive moving average arma models with unspecified innovation densities the finite sample performances of these procedures are investigated here in the context of ar order identification and compared to those of classical partial correlograms and lagrange multipliers methods the results achieved by rank based methods are quite comparable in the gaussian case to those achieved by the traditional ones which under gaussian assumptions are asymptotically optimal however under non gaussian innovation densities especially heavy tailed or nonsymmetric or when outliers are present the percentages of correct order selection based on rank methods are strikingly better than those resulting from traditional approaches even in the case of very short n 25 series these empirical findings confirm the often ignored theoretical fact that the gaussian case in the arma context

a bayes procedure for the identification of univariate time series models on presente un critere d identification qui est asymptotiquement equivalent a une regle de decision bayesienne

comment on adaptive estimation in time series regression models by d g steigerwald abstract in this comment we discuss some serious shortcomings and errors in steigerwald 1992

adaptive estimation in time series regression models this work develops adaptive estimators for a linear regression model with serially correlated errors we show that these results continue to hold when the order of the arma process characterizing the errors is unknown the finite sample results are promising indicating that substantial efficiency gains may be possible for samples as small as 50 observations we use these estimators to investigate the behavior of the forward foreign exchange market

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

diagnostic tests for multiple time series models this paper is concerned with the development and application of diagnostic checks for vector linear time series models a hypothesis testing procedure based upon the score or lagrangean multiplier principle is advocated and the distributions of the test statistic both under the null hypothesis and under a pitman sequence of alternatives are discussed consideration of alternative models with singular sensitivity matrices when the null hypothesis is true leads to an interpretation of the score test as a pure significance test and to a notion of an equivalence class of local alternatives portmanteau tests of model adequacy are also investigated and are seen to be equivalent to score tests

consistent moment selection procedures for generalized method of moments estimation this paper considers a generalized method of moments gmm estimation problem in which one has a vector of moment conditions some of which are correct and some incorrect the paper introduces several procedures for consistently selecting the correct moment conditions the procedures also can consistently determine whether there is a sufficient number of correct moment conditions to identify the unknown parameters of interest the paper specifies moment selection criteria that are gmm analogues of the widely used bic and aic model selection criteria the latter is not consistent the paper also considers downward and upward testing procedures all of the moment selection procedures discussed in this paper are based on the minimized values of the gmm criterion function for different vectors of moment conditions the procedures are applicable in time series and cross sectional contexts application of the results of the paper to instrumental variables estimation problems yields consistent procedures for selecting instrumental variables

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

local rank tests in a multivariate nonparametric relationship abstract consider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables for a fixed level of one set of explanatory variables we provide consistent statistical tests called local rank tests to determine whether the multivariate relationship can be explained by a smaller number of functions we also provide estimators for the smallest number of functions called local rank explaining the relationship the local rank tests and the estimators of local rank are defined in terms of the eigenvalues of a kernel based estimator of some matrix the asymptotics of the eigenvalues is established by using the so called fujikoshi expansion along with some techniques of the theory of u statistics we present a simulation study which examines the small sample properties of local rank tests we also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set this work can be viewed as a local extension of the tests for a number of factors in a nonparametric relationship introduced by stephen donald

the properties of the parameterization of armax systems and their relevance for structural estimation and dynamic specification armax systems in structural specification are considered topological and geometric properties of the parameter space and of the parameterization which are important for the properties of the estimators regardless of their special form are investigated and the specification of the maximum lag lengths is discussed

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

chaos and the approach to equilibrium in a discrete sine gordon equation abstract in this paper we study the long time dynamics of a hamiltonian system with many degrees of freedom we numerically investigate the system s approach to equipartation of energy when the intial energy is confined to one or a small set of fourier modes we find that there is a transition from a low energy regime in which energy does not spread appreciably among the modes to a high energy regime in which the system rapidly approaches equipartition and that this transition coincides with the onset of chaos in the system as evidenced by a sharp rise in the largest lyapunov exponent for low frequency initial conditions the critical parameter for this transition is the scaled energy e1 l n 2e using a generalization of the traditional chirikov resonance overlap calculation on a three mode subset of the full system we predict the onset of widespread chaos and the transition to equipartition on relatively short timescales

consistent model and moment selection procedures for gmm estimation with application to dynamic panel data models this paper develops consistent model and moment selection criteria for gmm estimation the criteria select the correct model specification and all correct moment conditions asymptotically the selection criteria resemble the widely used likelihood based selection criteria bic hqic and aic the latter is not consistent the gmm selection criteria are based on the j statistic for testing over identifying restrictions bonus terms reward the use of fewer parameters for a given number of moment conditions and the use of more moment conditions for a given number of parameters the paper also considers a consistent downward testing procedure the paper applies the model and moment selection criteria to dynamic panel data models with unobserved individual effects the paper shows how to apply the selection criteria to select the lag length for lagged dependent variables to detect the number and locations of structural breaks to determine the exogeneity of regressors and or to determine the existence of correlation between some regressors and the individual effect to illustrate the finite sample performance of the selection criteria and the testing procedures and their impact on parameter estimation the paper reports the results of a monte carlo experiment on a dynamic panel data model

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

chaotic transport and integrable instabilities in a near integrable many particle hamiltonian lattice abstract this paper is a report of detailed numerical experiments on a large dimensional hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine gordon equation through four different one parameter families of initial data we focus on the interplay between numerically observable near integrable behavior quantified by modern statistical diagnostics and the integrable pde solutions and phase space geometry associated with the initial data transitions are observed within each family from quasiperiodic behavior to chaotic motion which is sometimes accompanied by transport toward equipartition these transitions in the near integrable system are correlated with integrable features of the initial data in particular transitions from neutrally stable to exponentially unstable sine gordon solutions and associated homoclinic orbits geometrically these correspond to transitions from neutrally stable tori to whiskered tori our results indicate that 1 positive lyapunov exponents our messure of chaos in near integrable orbits coincide one to one with integrable pde instabilities and homoclinic orbits 2 the statistical transition to equipartition of energy on a long fixed timescale occurs only if but not necessarily if there are integrable instabilities and associated whiskered tori the rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom 3 isolated cases of discreteness induced instabilities and action transport are observed these return to quasiperiodic flow at finer discretizations 4 a simple measure such as energy of the initial data is not an accurate predictor of chaotic or recurrent behavior one family of initial data exhibits as the energy increases alternating windows of quasiperiodic and chaotic orbits

spectral decomposition of bell s operators for qubits the spectral decomposition is given for the n qubit bell operators with two observables per qubit it is found that the eigenstates when non degenerate are n qubit ghz states even for those operators that do not allow the maximal violation of the corresponding inequality we present two applications of this analysis in particular we discuss the existence of pure entangled states that do not violate the mermin klyshko inequality for n 3

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

isospectral flow in loop algebras and quasiperiodic solutions of the sine gordon equation the sine gordon equation is considered in the hamiltonian framework provided by the adler kostant symes theorem the phase space a finite dimensional coadjoint orbit in the dual space g of a loop algebra g is parameterized by a finite dimensional symplectic vector space w embedded into g by a moment map real quasiperiodic solutions are computed in terms of theta functions using a liouville generating function which generates a canonical transformation to linear coordinates on the jacobi variety of a suitable hyperelliptic curve

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

inference concerning the number of factors in a multivariate nonparametric relationship this paper considers the problem of determining the number of nonparametric factors in a multivariate nonparametric relationship the definition given is broad enough to encompass a number of potential applications in econometrics including inferring the rank of demand testing whether for a given set of instruments it is possible to identify a linear model and testing arbitrage pricing theory the paper gives methods for testing hypotheses concerning the number of factors using series and kernel based nonparametric methods and also considers the problem of estimating the number of factors the methods are compared in a small simulation study and an application to determining the rank of demand systems is considered

autoresonant excitation of multiphase waves in the sine gordon model abstract autoresonant excitation and control of multiphase waves of the sine gordon equation is discussed the autoresonance a persistent nonlinear phase locking is achieved by applying chirped frequency wave like perturbations and passage through resonances with inactive dormant degrees of freedom in the system successful phase locking requires the amplitude of the driving perturbation to exceed a threshold that scales as 3 4 power of the driving frequency chirp rate the spectral theory of the inverse scattering transform method is used for diagnostics and analysis of the excited solutions

structure identification of parameter bounding models by use of noise structure bounds the selection of a model for a given set of records is discussed for the case when the output error is required to be always within some specified bounds a new selection criterion is proposed the model should be the simplest giving sufficiently unstructured output errors the lack of structure is specified by requiring the candidate model to be capable of giving sufficiently small sample autocorrelations of the model output errors over a range of lags computational algorithms to apply the criterion are presented and illustrated by tests on ar processes

inferring the rank of a matrix abstract this paper considers methods of inference concerning the rank of matrix a based on an asymptotically normal estimate of and some identifiable specification for one such specification is 0 in which case one is interested in the rank of we first propose and examine the properties of a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger we then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing conditions for consistency of such procedures are obtained some economic applications are discussed and the various methods are compared in a monte carlo experiment

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system