non self adjoint zakharov shabat operator with a potential of the finite asymptotic values i direct spectral and scattering problems the zakharov and shabat equation for the scattering problem is studied the estimates analytical properties and asymptotic expansions of the jost solution are presented for a general class of the potentials q x not vanishing at infinity the existence of the similarity transformation is also shown for q x vanishing at infinity the continuous part of the spectrum doubly degenerates however nonvanishing finite asymptotic values of q x dissolve the degeneracy completely the expansion theorem is given in c 0 2 r and for a class of q x we prove that the zakharov and shabat equation yields a non self adjoint spectral operator in the hilbert space in the sense of dunford and schwartz

integrable nonlocal wave interaction models the equations we investigate describe the nonlinear and nonlocal interaction of waves in 1 1 dimensions with potential applicability although these equations are integrable the solution of the initial and boundary value problem requires a judicious use of spectral methods in particular we show that spectral data evolve linearly only for special boundary values

bounded ultra elliptic solutions of the defocusing nonlinear schrodinger equation abstract an effective integration method is presented for the bounded ultra elliptic solutions of the defocusing nonlinear schrodinger equation the two phase solutions are explicitly parametrized in terms of two physically meaningful variables the energy density and the momentum density cavitation viz a minimum amplitude of zero occurs if and only if the length of the largest spectral band is less than or equal to the sum of the lengths of the two smaller spectral bands in the case of strict inequality there are exactly two cavitation points in each period parallelogram

solution of the cauchy problem for a generalized sine gordon equation we consider a spectral problem generating a hierarchy of nonlinear evolution equations including the sine gordon equation and a physically interesting generalization in the laboratory coordinates the direct and inverse problems are treated the time evolution of the spectral data is explicitly given and therefore the cauchy problem for the related equations is solved

spectral transform and solitons for the three wave coupling model with nontrivial boundary conditions a nonlinear three wave coupling model with nontrivial boundary conditions is analyzed in the framework of the inverse spectral transform spectral data are determined and their evolution is derived it is shown that there exists mutual cancellation of divergences of the scattering matrix elements in branching points on the plane of a spectral parameter a regular dark soliton type solution is obtained

distribution free bounds for serial correlation coefficients in heteroskedastic symmetric time series abstract we consider the problem of testing whether the observations x 1 x n of a time series are independent with unspecified possibly nonidentical distributions symmetric about a common known median various bounds on the distributions of serial correlation coefficients are proposed exponential bounds eaton type bounds chebyshev bounds and berry esseen zolotarev bounds the bounds are exact in finite samples distribution free and easy to compute the performance of the bounds is evaluated and compared with traditional serial dependence tests in a simulation experiment the procedures proposed are applied to u s data on interest rates commercial paper rate

on bootstrap inconsistency and bonferroni based size correction for the subset anderson rubin test under conditional homoskedasticity abstract we focus on the linear instrumental variable model with two endogenous regressors under conditional homoskedasticity and study the subset anderson and rubin 1949 ar test when the nuisance structural parameter the unrestricted slope coefficient of endogenous regressor may be weakly identified weak identification leads to nonstandard null limiting distributions and alternative to the usual chi squared critical value is needed we first investigate the bootstrap validity for the subset ar test based on various plug in estimators and show that the bootstrap provides asymptotic refinement when the nuisance structural parameter is strongly identified but is inconsistent when it is weakly identified this is in contrast to the result of bootstrap validity in moreira et al 2009 then we propose a bonferroni based size correction method that yields correct asymptotic size for all the test statistics considered the power performance of size corrected tests can be further improved by applying the mapping between structural and endogenous parameters in the model monte carlo experiments confirm the bootstrap inconsistency and demonstrate that all the subset tests based on our correction technique control the size

exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions this paper proposes finite sample procedures for testing the sure specification in multi equation regression models i e whether the disturbances in different equations are contemporaneously uncorrelated or not we apply the technique of monte carlo mc tests see dwass and barnard respectively ann math statist 28 1957 181 j r statist soc ser b 25 1963 294 to obtain exact tests based on standard lr and lm zero correlation tests we also suggest a mc quasi lr qlr test based on feasible generalized least squares fgls we show that the latter statistics are pivotal under the null which provides the justification for applying mc tests furthermore we extend the exact independence test proposed by harvey and phillips bull econ res 34 2 1982 79 to the multi equation framework specifically we introduce several induced tests based on a set of simultaneous harvey phillips type tests and suggest a simulation based solution to the associated combination problem the properties of the proposed tests are studied in a monte carlo experiment which shows that standard asymptotic tests exhibit important size distortions while mc tests achieve complete size control and display good power moreover mc qlr tests performed best in terms of power a result of interest from the point of view of simulation based tests the power of the mc induced tests improves appreciably in comparison to standard bonferroni tests and in certain cases outperform the likelihood based mc tests the tests are applied to data used by fischer j monetary econ 32 485 1993 to analyze the macroeconomic determinants of growth

soliton solutions of the kadomtsev petviashvili ii equation we study a general class of line soliton solutions of the kadomtsev petviashvili ii kpii equation by investigating the wronskian form of its tau function we show that in addition to the previously known line soliton solutions of kpii this class also contains a large variety of multisoliton solutions many of which exhibit nontrivial spatial interaction patterns we also show that in general such solutions consist of unequal numbers of incoming and outgoing line solitons from the asymptotic analysis of the tau function we explicitly characterize the incoming and outgoing line solitons of this class of solutions we illustrate these results by discussing several examples

exact non parametric tests for a random walk with unknown drift under conditional heteroscedasticity this paper proposes a class of linear signed rank statistics to test for a random walk with unknown drift in the presence of arbitrary forms of conditional heteroscedasticity the class considered includes analogues of the well known sign and wilcoxon test statistics the exactness of the proposed tests rests only on the assumption that the errors are symmetrically distributed no other assumptions such as normality or even the existence of moments are required these tests extend the non parametric approach to testing for a random walk proposed in campbell and dufour international economic review 38 1997 151 simulations confirm the reliability of the new tests and their power can be considerably superior to the bounds tests of campbell and dufour especially for alternatives close to the null the inference methods developed are illustrated by a test of the random walk hypothesis in exchange rates for five major currencies against the u s dollar

mmc techniques for limited dependent variables models implementation by the branch and bound algorithm we propose a finite sample approach to some of the most common limited dependent variables models the method rests on the maximized monte carlo mmc test technique proposed by dufour 1998 monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics journal of econometrics this issue we provide a general way for implementing tests and confidence regions we show that the decision rule associated with a mmc test may be written as a mixed integer programming problem the branch and bound algorithm yields a global maximum in finite time an appropriate choice of the statistic yields a consistent test while fulfilling the level constraint for any sample size the technique is illustrated with numerical data for the logit model c 2005 elsevier b v all rights reserved

bright n soliton solutions in terms of the triple wronskian for the coupled nonlinear schrodinger equations in optical fibers the coupled nonlinear schrodinger nls equations are usually used to describe the dynamics of two component solitons in optical fibers via the darboux transformation the coupled nls equations of the manakov type are found to have triple wronskian solutions proof is finished by virtue of some new triple wronskian identities by solving the zero potential lax pair the triple wronskian solutions give the bright n soliton solutions which are characterized by 3n complex parameters to obtain an understanding of the asymptotic behavior of the bright n soliton solutions with arbitrary n some algebraic properties of the triple wronskian are analyzed and an algebraic procedure is presented to derive the explicit expressions of the asymptotic solitons as t

an extension of the wronskian technique for the multicomponent wronskian solution to the vector nonlinear schrodinger equation in this paper the wronskian technique is applied to the vector nonlinear schrodinger equation with arbitrary m components which arises from some applications in the multimode fibers photorefractive materials and bose einstein condensates via the iterative algorithm based on the darboux transformation the m 1 component wronskian solution is generated from the zero solution the verification of the solution is finished by using the m 1 component wronskian notation and new determinantal identities with a set of n linearly independent solutions of the zero potential lax pair the m 1 component wronskian solution is found to be the representation of the bright n soliton solution which contains m 1 n parameters for characterizing the asymptotic behavior of the generic bright n soliton solution an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t

exact tests and confidence sets in linear regressions with autocorrelated errors this article proposes a general method to build exact tests and confidence sets in linear regressions with first order autoregressive gaussian disturbances because of a nuisance parameter problem we argue that generalized bounds tests and conservative confidence sets provide natural inference procedures in such a context given an exact confidence set for the autocorrelation coefficient we describe how to obtain a similar simultaneous confidence set for the autocorrelation coefficient and any subvector of regression coefficient conservative confidence sets for the regression coefficients are then deduced by a projection method for any hypothesis that specifies jointly the value of the autocorrelation coefficient and any set of linear restrictions on the regression coefficients we get exact similar tests for tesing linear hypotheses about the regression coefficients only we suggest bounds type procedures exact confidence sets for the autocorrelation coefficient are built by inverting autocorrelation tests the method is illustrated with two examples copyright 1990 by the econometric society

higher order constrained hamiltonian systems in this paper we study a hamiltonian formulation of the higher order constrained systems hocss defined previously in lagrangian terms we shall focus on second order constraints in positions i e second order hocss which correspond on the hamiltonian side to first order constraints in momenta we show that for affine second order constraints the existence and uniqueness conditions can be stated for their corresponding equations of motion and that the latter can be written in terms of a leibniz bracket we also show that hamiltonian systems with external forces can be described as suitable affine second order hocss we study some examples as systems with friction and we present an application for the global asymptotic stabilization of servomechanisms

further results on projection based inference in iv regressions with weak collinear or missing instruments we study a general family of anderson rubin type procedures allowing for arbitrary collinearity among the instruments and endogenous variables using finite sample distributional theory we show that the proposed procedures besides being robust to weak instruments are also robust to the exclusion of relevant instruments and to the distribution of endogenous regressors a solution to the problem of computing linear projections from general possibly singular quadric surfaces is derived and used to build finite sample confidence sets for individual structural parameters the importance of robustness to excluded instruments is studied by simulation applications to the trade growth relationship and to education returns are presented

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

modulational instability and exact solutions of the nonlinear schrodinger equation coupled with the nonlinear klein gordon equation the modulational instability mi of the coupled nonlinear schrodinger and nonlinear klein gordon equations is investigated it is found that there are a number of possibilities for the mi regions due to the generalized dispersion relation which relates the frequency and wavenumber of the modulating perturbations some exact travelling wave solutions are constructed via the solutions of a 4 model through a simple mapping relation furthermore we present five different types of solutions representing possible final states of modulationally unstable perturbations the profiles of solitary wave structures are displayed for some fixed parameters

integrable models in nonlinear optics and soliton solutions nonlinear optics more than other subjects has surprisingly produced quite a number of approximate models which are integrable here we briefly mention these models with the aim of showing contacts between the mathematical and applicative sides of the theory of integrable dynamical systems in particular we illustrate the recent impact of boomeronic type wave equations on applicative issues in the special context of the resonant interaction of three waves

inverse scattering on the line for a dirac system the whole line version of the gelfand levitan marchenko glm equation for a dirac system is studied a new derivation of the glm equation is given under weaker hypotheses than frolov s earlier treatment sov math dokl 13 1468 1972 and the complete inversion is carried out in some explicit cases in which a spectral gap is present previous calculations of this type are restricted either to a scalar potential or degenerate gap applications are discussed in connection with optical couplers and soliton equations

solution of three waves interaction type models with nontrivial asymptotic and boundary conditions in the following we introduce a new way to solve the cauchy problem for non vanishing potentials at infinity for the spectral problem of the zakharov and shabat ablowitz kaup newell and segur zs akns hierarchy then we apply the results to solve a class of coupled systems with this nonvanishing condition on the potential and with quite general boundary conditions on the other fields finally we present an application to a physical system governed by a three waves interaction type model

inverse scattering transform of an extended nonlinear schrodinger equation with nonzero boundary conditions and its multisoliton solutions abstract under investigation in this work is an extended nonlinear schrodinger equation with nonzero boundary conditions which can model the propagation of waves in dispersive media firstly a matrix riemann hilbert problem for the equation with nonzero boundary conditions at infinity is systematically discussed then the inverse problems are solved through the investigation of the matrix riemann hilbert problem therefore the general solutions for the potentials and explicit expressions for the reflection less potentials are presented furthermore we construct the simple pole and double pole solutions for the equation finally the remarkable characteristics of these solutions are graphically discussed our results should be useful to enrich and explain the related nonlinear wave phenomena in nonlinear fields

discrete solitons of the focusing ablowitz ladik equation with nonzero boundary conditions via inverse scattering soliton solutions of the focusing ablowitz ladik equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform ist after reviewing the relevant aspects of the direct and inverse problems explicit soliton solutions are obtained which are the discrete analog of the tajiri watanabe and kuznetsov ma solutions to the focusing nonlinear schrodinger equation on a finite background then by performing suitable limits of the above solutions discrete analogs of the celebrated akhmediev and peregrine solutions are also presented the latter which can be thought of as a discrete rogue wave is expressed as a family of rational functions of the discrete spatial variable n and time parametrically depending on the amplitude qo of the background these solutions which had been recently derived by direct methods are obtained for the first time within the framework of the ist thus also providing a spectral characterization of the solutions

integrability of nonlinear wave equations and solvability of their initial value problem we investigate one particular prototypical system of evolution equations in 1 1 dimensions which models the nonlocal interaction of two waves in quadratic nonlinear media these equations are integrable as they possess a lax pair we consider the spectral method of solving the associated initial value problem and as recently pointed out we show that generically the associated initial value problem cannot be linearized we further elaborate on the integrability issue for this model by displaying the corresponding conservation laws and the motion of a particular soliton

construction of new solutions to the fully nonlinear generalized camassa holm equations by an indirect f function method an indirect f function method is introduced to solve the generalized camassa holm equation with fully nonlinear dispersion and fully nonlinear convection c l n p taking advantage of elliptic equation this f function is used to map the solutions of the generalized camassa holm equation to those of the elliptic equation as a result we can successfully obtain in a unified way and for special values of the parameters of this equation many exact solutions expressed by various single and combined nondegenerative jacobi elliptic function solutions and their degenerative solutions soliton combined soliton solutions and triangular solutions as the modulus m is driven to 1 and 0

purely imaginary eigenvalues of zakharov shabat systems zakharov shabat systems with single hump and real but not necessarily symmetric potentials are shown to have purely imaginary eigenvalues only coupled with examples of double hump potentials with nonimaginary eigenvalues this establishes that confinement of zakharov shabat eigenvalues to the imaginary axis is a characteristic of potentials whose energy is concentrated in a single region of the time axis

the vector nonlinear schrodinger hierarchy abstract an article published recently by one of the authors jne presents closed form solutions for zero curvature representations of the vector nonlinear schrodinger hierarchy several of these results are confirmed computationally but left unproven in this article we begin by providing strict algebraic proofs of these results the forms of the hierarchy s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants we admit specific non zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner thereby accessing the full family of finite gap solutions to the vector nonlinear schrodinger equation finally an original construction of the infinite set of conserved densities of the hierarchy is given

inverse scattering transform for the focusing nonlinear schrodinger equation with nonzero boundary conditions the inverse scattering transform for the focusing nonlinear schrodinger equation with non zero boundary conditions at infinity is presented including the determination of the analyticity of the scattering eigenfunctions the introduction of the appropriate riemann surface and uniformization variable the symmetries discrete spectrum asymptotics trace formulae and the so called theta condition and the formulation of the inverse problem in terms of a riemann hilbert problem in addition the general behavior of the soliton solutions is discussed as well as the reductions to all special cases previously discussed in the literature

a degeneration of two phase solutions of the focusing nonlinear schrodinger equation via riemann hilbert problems two phase solutions of focusing nls equation are classically constructed out of an appropriate riemann surface of genus two and expressed in terms of the corresponding theta function we show here that in a certain limiting regime such solutions reduce to some elementary ones called solitons on unstable condensate this degeneration turns out to be conveniently studied by means of basic tools from the theory of riemann hilbert problems in particular no acquaintance with riemann surfaces and theta function is required for such analysis

exact soliton solutions of coupled nonlinear schrodinger equations shape changing collisions logic gates and partially coherent solitons the different dynamical features underlying soliton interactions in coupled nonlinear schrodinger equations which model multimode wave propagation under varied physical situations in nonlinear optics are studied in this paper by explicitly constructing multisoliton solutions up to four soliton solutions for two coupled and arbitrary n coupled nonlinear schrodinger equations using the hirota bilinearization method we bring out clearly the various features underlying the fascinating shape changing intensity redistribution collisions of solitons including changes in amplitudes phases and relative separation distances and the very many possibilities of energy redistributions among the modes of solitons however in this multisoliton collision process the pairwise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons detailed asymptotic analysis also shows that when solitons undergo multiple collisions there exists the exciting possibility of shape restoration of at least one soliton during interactions of more than two solitons represented by three and higher order soliton solutions from an application point of view we have shown from the asymptotic expressions how the amplitude intensity redistribution can be written as a generalized linear fractional transformation for the n component case also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters leading to possible multistate logic in addition we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape changing collisions thereby explaining their variable profile and shape variation in collision process

the dirac equation and integrable systems of kp type the propagator for the 2d heat equation in an arbitrary linear space is shown to give solutions of the two component kadomtsev petviashvilii kp equations also called davey stewartson system this propagator is subject to the klein gordon equation and its right derivatives are required to be of rank one that imply that it can be expressed in terms of solutions of the dirac equation large families of solutions of the two component kadomtsev petviashvilii equations are constructed in terms of solutions of the heat and dirac equations particular attention is paid to the real reductions of the davey stewartson type recovering in this way the line solitons and the multidromion solutions moreover new solutions to the davey stewartson i are presented as massive deformations of the dromion

a hierarchy of coupled evolution equations with self consistent sources and the dressing method the dressing method based on the local 3 3 matrix problem is used to study the two coupled integrable nonlinear evolution equations with self consistent sources a hierarchy of the nonlinear evolution equations associated with a 3 3 matrix problem including the coupled nonlinear schr dinger equation and maxwell bloch equation is derived by introducing a suitable recursion operator explicit solutions including n soliton solutions for the hierarchy in a closed form are obtained by virtue of the properties of the cauchy matrix

inverse scattering transform for the vector nonlinear schrodinger equation with nonvanishing boundary conditions the inverse scattering transform for the vector defocusing nonlinear schrodinger nls equation with nonvanishing boundary values at infinity is constructed the direct scattering problem is formulated on a two sheeted covering of the complex plane two out of the six jost eigenfunctions however do not admit an analytic extension on either sheet of the riemann surface therefore a suitable modification of both the direct and the inverse problem formulations is necessary on the direct side this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions the discrete spectrum bound states and symmetries of the direct problem are then discussed in the most general situation a discrete eigenvalue corresponds to a quartet of zeros poles of certain scattering data the inverse scattering problem is formulated in terms of a generalized riemann hilbert rh problem in the upper lower half planes of a suitable uniformization variable special soliton solutions are constructed from the poles in the rh problem and include dark dark soliton solutions which have dark solitonic behavior in both components as well as dark bright soliton solutions which have one dark and one bright component the linear limit is obtained from the rh problem and is shown to correspond to the fourier transform solution obtained from the linearized vector nls system

new integrable pdes of boomeronic type novel integrable systems of coupled first order autonomous pdes in 1 1 dimensions space x and time t are presented integrable covariant 2 vector and 3 vector pdes as well as higher order integrable pdes for a single or a couple of scalar dependent variables including an extension of the sine gordon equation and a remarkably neat highly nonlinear third order pde are also obtained by appropriate reductions of the basic matrix equations the lax pairs that characterize the integrable character of the basic matrix pdes are also exhibited as well as their single soliton solutions these solitons generally exhibit the boomeronic and less generically the trapponic phenomenology namely they do not move uniformly but rather in an appropriate reference system come in from one end in the remote past and boomerang back to that same end in the remote future boomerons or are trapped to oscillate around a value fixed by the initial data trappons

rational solitons of wave resonant interaction models integrable models of resonant interaction of two or more waves in 1 1 dimensions are known to be of applicative interest in several areas here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear schrodinger equations and the equations describing the resonant interaction of three waves the darboux dressing construction of soliton solutions is applied under the condition that the solutions have rational or mixed rational exponential dependence on coordinates our algebraic construction relies on the use of nilpotent matrices and their jordan form we systematically search for all bounded rational mixed rational exponential solutions and find a broad family of such solutions of the three wave resonant interaction equations

exact tests for structural change in first order dynamic models abstract several finite sample tests of parameter constancy against the presence of structural change are proposed for a linear regression model with one lagged dependent variable and independent normal disturbances the procedures derived include analysis of covariance cusum cusum of squares and predictive tests the approach used to obtain the tests involves the application of three techniques derivation of an exact confidence set for the autoregressive parameter based on using an appropriately extended regression a union intersection technique and when required randomization the tests proposed are illustrated with some artificial data and applied to a dynamic trend model of gross private domestic investment in the u s

casorati determinant solution for the relativistic toda lattice equation the relativistic toda lattice equation is decomposed into three toda systems the toda lattice itself backlund transformation of toda lattice and discrete time toda lattice it is shown that the solutions of the equation are given in terms of the casorati determinant by using the casoratian technique the bilinear equations of toda systems are reduced to the laplace expansion form for determinants the n soliton solution is explicitly constructed in the form of the casorati determinant

painleve analysis and integrability of coupled non linear schrodinger equations considering a system of coupled non linear schrodinger nls equations the authors discuss the integrability properties through painleve p analysis for the two coupled nls equations they show that there exists a pair of parametric values possessing the p property for which the associated backlund transformation bt and the hirota bilinearisation are constructed these parametric choices are identical to those of zakharov and schulman 1982 who established the integrability in terms of motion invariants finally they extend the p analysis to the n coupled nls system and identify two parametric choices possessing the p property which are natural generalisations of the two coupled nls cases

long time asymptotics for a mixed nonlinear schrodinger equation with the schwartz initial data abstract in this paper we define a general analytical domain and two reflection coefficients to study the long time asymptotics for a defocusing mixed nonlinear schrodinger equation with the schwartz initial data it is shown that the solution of the initial value problem for the mixed schrodinger equation can be expressed in terms of the solution associated with a riemann hilbert problem the long time asymptotic of the mixed schrodinger equation is further obtained via the deift zhou nonlinear steepest descent method the long time asymptotic for the classical schrodinger equation derivative schrodinger equation and modified schrodinger equation are obtained directly from our results as special cases

lagrangian systems with higher order constraints a class of mechanical systems subject to higher order constraints i e constraints involving higher order derivatives of the position of the system are studied we call them higher order constrained systems hocss they include simplified models of elastic rolling bodies and also the so called generalized nonholonomic systems gnhss whose constraints only involve the velocities of the system i e first order derivatives in the position of the system one of the features of this kind of systems is that d alembert s principle or its nonlinear higher order generalization the chetaev s principle is not necessarily satisfied we present here as another interesting example of hocs systems subjected to friction forces showing that those forces can be encoded in a second order kinematic constraint the main aim of the paper is to show that every hocs is equivalent to a gnhs with linear constraints in a canonical way that is to say systems with higher order constraints can be described in terms of one with linear constraints in velocities we illustrate this fact with a system with friction and with rocard s model dynamique generale des vibrations 1949 chap xv p 246 and l instabilite en mecanique automobiles avions ponts suspendus 1954 of a pneumatic tire as a by product we introduce some applications on higher order tangent bundles which we expect to be useful for the study of intrinsic aspects of the geometry of such bundles

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

a generalization of chetaev s principle for a class of higher order nonholonomic constraints the constraint distribution in nonholonomic mechanics has a double role on the one hand it is a kinematic constraint that is it is a restriction on the motion itself on the other hand it is also a restriction on the allowed variations when using d alembert s principle to derive the equations of motion we will show that many systems of physical interest where d alembert s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities this includes for example elastic rolling bodies and pneumatic tires also d alembert s principle and chetaev s principle fall into this scheme we emphasize the geometric point of view avoiding the use of local coordinates which is the appropriate setting for dealing with questions of global nature like reduction

monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics abstract the technique of monte carlo mc tests dwass 1957 annals of mathematical statistics 28 181 187 barnard 1963 journal of the royal statistical society series b 25 294 provides a simple method for building exact tests from statistics whose finite sample distribution is intractable but can be simulated when no nuisance parameter is involved we extend this method in two ways first by allowing for mc tests based on exchangeable possibly discrete test statistics second by generalizing it to statistics whose null distribution involves nuisance parameters maximized mc mmc tests simplified asymptotically justified versions of the mmc method are also proposed these provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics

joint one sided tests of linear regression coefficients abstract the main example of the class of problems considered below is that of testing whether a subset of regression coefficients are jointly zero assuming knowledge of the coefficients signs if this knowledge is ignored the likelihood ratio wald and lagrange multiplier tests are each equivalent to the f test we propose a new test which can be applied as a one sided t test and which is umpi in a subspace of the parameter space empirical power comparisons with the power envelope the f test and the exact one sided likelihood ratio test show that the new test can have exceptionally good power over a wide range of the parameter space

simulation based finite sample tests for heteroskedasticity and arch effects tests for heteroskedasticity in linear regressions are typically based on asymptotic approximations we show that the size of such tests can be perfectly controlledin nite samples through monte carlo test techniques with both gaussian and non gaussian heavy tailed disturbance distributions the procedures studied include standard heteroskedasticity tests e g glejser bartlett cochran hartley breusch pagan godfrey white szroeter as well as tests for arch type heterosked asticity sup type andcombinedtests are also proposedto allow for unknown breakpoints in the variance the fact that the proposedproced ures achieve size control andhave goodpower is demonstrated in a monte carlo simulation c 2003 elsevier b v all rights reserved

on the backlund gauge transformation and homoclinic orbits of a coupled nonlinear schrodinger system abstract the backlund gauge transformation for a system of coupled nls nonlinear schrodinger equations with a degenerate associated spectral operator is derived from an algebraic perspective extending aspects of other results m boiti tu guizhang il nuovo cimento 71b 1982 253 264 d h sattinger v d zurkowski physica d 26 1 3 1987 225 250 that apply in the context of non degenerate spectral operators moreover we demonstrate how the backlund gauge transformation can be used to explicitly construct the entire unstable manifold via superpositions of homoclinic orbits of a plane wave solution with both self phase instabilities and coupling instabilities this work builds on the results of ercolani et al n ercolani m g forest d w mclaughlin physica d 18 1986 472 474 n ercolani m g forest d w mclaughlin physica d 43 2 3 1990 349 384 for the sine gordon equation and forest et al m g forest d w mclaughlin d j muraki o c wright j nonlinear sci in press m g forest s p sheu o c wright phys lett a in press for the integrable coupled nls system

an explicit expression of the dark n soliton solution of the mkdv equation by means of the darboux transformation a darboux transformation is developed for generating dark multi soliton solutions of the mkdv equation the darboux matrices are found explicitly in recursive manner and a system of linear algebraic equations is derived for determining the dark n soliton solutions by means of the binet cauchy formula an explicit expression of the dark n soliton solution is obtained

on the sensitivity of the usual t and f tests to covariance misspecification we consider the standard linear regression model with all standard assumptions except that the disturbances are not white noise but distributed n 0 p2x h where x 0 i n our interest lies in testing linear restrictions using the usual f statistic based on ols residuals we are not interested in nding out whether h 0 or not instead we want to nd out what the e ect is of possibly nonzero h on the f statistic itself we propose a sensitivity statistic for this purpose discuss its distribution and obtain a practical and easy to use decision rule to decide whether the f test is sensitive or not to covariance misspeci cation when h is close to zero some nite and asymptotic properties of u are studied as well as its behaviour in the special case of an ar 1 process near the unit root 2000 elsevier science s a all rights reserved jel classixcation c12 c22 c51 c52

the alternative durbin watson test an assessment of durbin and watson s choice of test statistic abstract this paper examines durbin and watson s 1950 choice of test statistic for their test of first 0order autoregressive regression disturbances attention is focused on an alternative statistic d theoretical and empirical power properties of the d test are compared with those of the durbin watson test the former is found to be locally best invariant while the latter is approximately locally best invariant the d test is also found to be more powerful than its counterpart against negative autocorrelation and for small values of the autocorrelation coefficient against positive autocorrelation selected bounds for significance points of d are tabulated

soliton solutions to the 3rd nonisospectral akns system abstract bilinear form of the nonisospectral akns equation is given the n soliton solutions and double wronskian solution are obtained through hirota s direct method and wronskian technique respectively the nonisospectral mkdv equation and its multi soliton solutions are presented by reducing

exact tests in single equation autoregressive distributed lag models abstract for hypotheses on the coefficient values of the lagged dependent variables in the arx class of dynamic regression models test procedures are developed which yield exact inference for given up to an unknown scale factor distribution of the innovation errors they include exact tests on the maximum lag length for structural change and on the presence of seasonal or multiple unit roots i e they cover situations where usually asymptotic and non exact t f aoc adf or hegy tests are employed the various procedures are demonstrated and compared in illustrative empirical models and the approach is critically discussed

inverse scattering transform for the nonlocal nonlinear schrodinger equation with nonzero boundary conditions in 2013 a new nonlocal symmetry reduction of the well known akns an integrable system of partial differential equations introduced by and named after mark j ablowitz david j kaup and alan c newell et al 1974 scattering problem was found it was shown to give rise to a new nonlocal pt symmetric and integrable hamiltonian nonlinear schrodinger nls equation subsequently the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized time periodic one soliton solutions was found in this paper the inverse scattering transform for the nonlocal nls equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes the direct and inverse scattering problems are analyzed specifically the direct problem is formulated the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained the inverse scattering problem which

markovian processes two sided autoregressions and finite sample inference for stationary and nonstationary autoregressive processes in this paper we develop finite sample inference procedures for stationary and non stationary autoregressive ar models the method is based on special properties of markov processes and a split sample technique the results on markovian processes intercalary independence and truncation only require the existence of conditional densities they are proved for possibly nonstationary and or non gaussian multivariate markov processes in the context of a linear regression model with ar 1 errors we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations this transformation leads to a new model which has the form of a two sided autoregression to which standard classical linear regression inference techniques can be applied we show how to derive tests and confidence sets for the mean and or autoregressive parameters of the model we also develop a test on the order of an autoregression we show that a combination of subsample based inferences can improve the performance of the procedure an application to u s domestic investment data illustrates the method

non self adjoint zakharov shabat operator with a potential of the finite asymptotic values ii inverse problem the inverse spectral and scattering problem of the zakharov shabat zs operator is studied the similarity transformation between zs operators is examined when their potentials q x have the common nonvanishing asymptotic values q at the infinity the marchenko equation is derived from the parseval equation we give the necessary as well as the sufficient condition of the scattering data for the potential of the specified class

the integrable nature of modulational instability we investigate the nonlinear stage of the modulational or benjamin feir instability by characterizing the initial value problem for the focusing nonlinear schrodinger nls equation with nonzero boundary conditions nzbc at infinity we do so using the recently formulated inverse scattering transform ist for this problem while the linearization of the nls equation ceases to be valid when the perturbations have grown sufficiently large compared to the background the results of the ist remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability we begin by studying the spectral problem for the dirac operator i e the first half of the lax pair for the nls equation with piecewise constant initial conditions which are a generalization to nzbc of a potential well and a potential barrier since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem the study of these kinds of poten

modulational instability and exact solutions for a three component system of vector nonlinear schrodinger equations the modulational instability mi of the three component system of vector nonlinear schrodinger equations is investigated it is found that there are a number of possibilities for the mi regions due to the generalized nonlinear dispersion relation which relates the frequency and the wave number of modulating perturbations some classes of exact traveling wave solutions are obtained under some special parameter values some representative wave structures are graphically displayed these solutions are obtained by the use of f expansion method

inverse scattering transform for the defocusing nonlinear schrodinger equation with fully asymmetric non zero boundary conditions abstract we formulate the inverse scattering transform ist for the defocusing nonlinear schrodinger nls equation with fully asymmetric non zero boundary conditions i e when the limiting values of the solution at space infinities have different non zero moduli the theory is formulated without making use of riemann surfaces and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem for the direct problem we give explicit single valued definitions of the jost eigenfunctions and scattering coefficients over the whole complex plane and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues we pose the inverse problem as a riemann hilbert problem on an open contour and we reduce the problem to a standard set of linear integral equations finally for comparison purposes we present the single sheet branch cut formulation of the inverse scattering transform for the initial value problem with symmetric equimodular non zero boundary conditions as well as for the initial value problem with one sided non zero boundary conditions and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts

some impossibility theorems in econometrics with applications to structural and dynamic models general characterizations of valid confidence sets and tests in problems which involve locally almost unidentified lau parameters are provided and applied to several econo metric models two types of inference problems are studied i inference about parame ters which are not identifiable on certain subsets of the parameter space and ii inference about parameter transformations with discontinuities when a lau parameter or parametric function has an unbounded range it is shown under general regularity conditions that any valid confidence set with level 1 a for this parameter must be unbounded with probability close to 1 a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model no valid confidence set which is almost surely bounded does exist these properties hold even if identifying restrictions are imposed similar results also obtain for parameters with bounded ranges consequently a confidence set which does not satisfy this characterization has zero coverage probability level this will be the case in particular for wald type confidence intervals based on asymptotic standard errors furthermore wald type statistics for testing given values of a lau parameter cannot be pivotal functions i e they have distributions which depend on unknown nuisance param eters and even cannot be usefully bounded over the space of the nuisance parameters these results are applied to several econometric problems inference in simultaneous equations instrumental variables iv regressions linear regressions with autoregressive errors inference about long run multipliers and cointegrating vectors for example it is shown that standard asymptotically justified confidence intervals based on iv estimators such as two stage least squares and the associated standard errors have zero coverage probability and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions a result of interest for interpreting iv regressions with weak instruments furthermore expansion methods e g edgeworth expansions and bootstrap techniques cannot solve these difficulties finally in a number of cases where wald type methods are fundamentally flawed e g iv regressions with poor instruments it is observed that likelihood based methods e g likelihood ratio tests and confidence sets combined with projection techniques can easily yield valid tests and confidence sets

darboux transformations for the nonlinear schrodinger equations darboux transformations for the akns zs system are constructed in terms of grammian type determinants of vector solutions of the associated lax pairs with an operator spectral parameter a study of the reduction of the darboux transformation for the nonlinear schrodinger equations with standard and anomalous dispersion is presented two different families of new solutions for a given seed solution of the nonlinear schrodinger equation are given being one family related to a new vector lax pair for it in the first family and associated to diagonal matrices we present topological solutions with different asymptotic argument for the amplitude and nonzero background for the anomalous dispersion case they represent continuous deformations of the bright n soliton solution which is recovered for zero background in particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear schrodinger equation associated with jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions the second family contains not only the solutions mentioned above but also broader classes of solutions for example in the standard dispersion case we are able to obtain the dark soliton solutions

uniform confidence bands characterization and optimality abstract this paper studies optimal uniform confidence bands for functions g x 0 where 0 is an unknown parameter vector we provide a simple characterization of a general class of taut 1 uniform confidence bands allowing for both nonlinear functions and nonparametrically estimated functions specifically we show that all taut bands can be obtained from projections on confidence sets for 0 and we characterize the class of sets which yield taut bands using these results we then present a computational method for selecting an approximately optimal confidence band for a given objective function we illustrate the applicability of these results in numerical applications

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

general n soliton solution to a vector nonlinear schrodinger equation we consider a general n soliton solution to a vector nonlinear schrodinger nls equation of all possible combinations of nonlinearities including all focusing all defocusing and mixed types based on the kp hierarchy reduction method we firstly construct general two bright one dark and one bright two dark soliton solutions in a three coupled nls equation then we extend our analysis to a vector nls equation to obtain a general n soliton solution in gram determinant form this formula unifies the bright dark and bright dark soliton solutions which have been widely studied in the literature the conditions for the existence of all types of soliton solutions with all possible combinations of nonlinearities are elucidated

projection based statistical inference in linear structural models with possibly weak instruments it is well known that standard asymptotic theory is not applicable or is very unreliable in models with identification problems or weak instruments one possible way out consists of using a variant of the anderson rubin 1949 ar procedure the latter allows one to build exact tests and confidence sets only for the full vector of the coefficients of the endogenous explanatory variables in a structural equation but not for individual coefficients this problem may in principle be overcome by using projection methods dufour 1997 dufour and jasiak 2001 at first sight however this technique requires the application of costly numerical algorithms in this paper we give a general necessary and sufficient condition that allows one to check whether an ar type confidence set is bounded furthermore we provide an analytic solution to the problem of building projection based confidence sets from ar type confidence sets the latter involves the geometric properties of quadrics and can be viewed as an extension of usual confidence intervals and ellipsoids only least squares techniques are needed to build the confidence intervals

multicomponent integrable wave equations i darboux dressing transformation the darboux dressing transformations are applied to the lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bright and dark soliton solutions the general formalism is set up and the relevant equations are explicitly solved several instances of multicomponent wave equations of applicative interest such as vector nonlinear schrodinger type equations and three resonant wave equations are considered

exponentially small asymptotics of solutions to the defocusing nonlinear schrodinger equation ii the riemann hilbert problem approach is used to derive a special set of leading order exponentially small asymptotics as t such that x t 0 of solutions to the cauchy problem for the defocusing nonlinear schrodinger equation i t u x2 u 2 u 2 1 u 0 with finite density initial data u x 0 x exp i 11 1 o 1 0 2

generalized nonholonomic mechanics servomechanisms and related brackets it is well known that nonholonomic systems obeying d alembert s principle are described on the hamiltonian side after using the legendre transformation by the so called almost poisson brackets in this paper we define the lagrangian and hamiltonian sides of a class of generalized nonholonomic systems gnhs obeying a generalized version of d alembert s principle such as rubber wheels like some simplified models of pneumatic tires and certain servomechanisms like the controlled inverted pendulum and show that corresponding equations of motion can also be described in terms of a bracket we present essentially all possible brackets in terms of which the mentioned equations can be written down which include the brackets that appear in the literature and point out those if any that are naturally related to each system in particular we show there always exists a leibniz bracket related to a gnhs and conversely that every leibniz system is a gnhs the control of the inverted pendulum on a cart

testing causality between two vectors in multivariate autoregressive moving average models abstract in the analysis of economic time series a question often raised is whether a vector of variables causes another one in the sense of granger most of the literature on this topic is concerned with bivariate relationships or uses finite order autoregressive specifications the purpose of this article is to develop a causality analysis in the sense of granger for general vector autoregressive moving average arma models we give a definition of granger noncausality between vectors which is a natural and simple extension of the notion of granger noncausality between two variables in our context this definition is shown to be equivalent to a more complex definition proposed by tjostheim for the class of linear invertible processes we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis this result is then specialized to the class of stationary invertible arma process

modulation instability the beginning abstract we discuss the early history of an important field of sturm and drang in modern theory of nonlinear waves it is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability a phenomenon resulting in a variety of nonlinear processes such as envelope solitons envelope shocks freak waves etc examples from water wave hydrodynamics electrodynamics nonlinear optics and convection theory are given

the inverse scattering transform for the focusing nonlinear schrodinger equation with asymmetric boundary conditions the inverse scattering transform ist as a tool to solve the initial value problem for the focusing nonlinear schrodinger nls equation with non zero boundary values ql r t al re 2ial r2t i l r as x is presented in the fully asymmetric case for both asymptotic amplitudes and phases i e with al ar and l r the direct problem is shown to be well defined for nls solutions q x t such that q x t ql r t l1 1 r with respect to x for all t 0 and the corresponding analyticity properties of eigenfunctions and scattering data are established the inverse scattering problem is formulated both via left and right marchenko integral equations and as a riemann hilbert problem on a single sheet of the scattering variables l r k2 al r2 where k is the usual complex scattering parameter in the ist the time evolution of the scattering coefficients is then derived showing that unlike the case of solutions with equal amplitudes as x here both reflection and transmission coefficients have