matrix kadomtsev petviashvili equation matrix identities and explicit non singular solutions a new version of the backlund darboux transformation for the matrix kadomtsev petviashvili kp equation is used to construct and study explicit multi parameter solutions and wavefunctions in terms of the matrix exponents a class of the self adjoint non singular solutions of kp i is introduced using the controllability notion from the system theory a subclass of the rationally decaying self adjoint non singular solutions is studied in particular several results prove new in the scalar case also

a second wronskian formulation of the boussinesq equation a wronskian formulation leading to rational solutions is presented for the boussinesq equation it involves third order linear partial differential equations whose representative systems are systematically solved the resulting solutions formulas provide a direct but powerful approach for constructing rational solutions positon solutions and complexiton solutions to the boussinesq equation various examples of exact solutions of those three kinds are computed the newly presented wronskian formulation is different from the one previously presented by li et al which does not yield rational solutions

the exact solution to boussinesq equation through a limiting procedure abstract it is shown that if a new bilinear backlund transformation to boussinesq equation has been reduced by the gauge transformation it always has another set of new n soliton solution by performing an appropriate limiting procedure on the already known ones

behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations we establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase showing that long time behavior is governed by an associated second order formal whitham modulation system a key point is to identify the way in which initial perturbations translate to initial data for this formal system a task accomplished by detailed estimates on the linearized solution operator about the background wave notably our approach gives both a common theoretical treatment and a complete classification in terms of phase coupling or decoupling of general systems of conservation or balance laws encompassing cases that had previously been studied separately or not at all at the same time our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase decoupled case further distinguishing behavior in the different cases an interesting technical aspect of our analysis is that for systems of conservation laws the whitham modulation description is of system rather than scalar form as a consequence of which renormalization methods such as have been used to treat the reaction diffusion case in general do not seem to apply

3d inverse scattering abstract we give a self contained presentation of the method developed by r newton for solving the 3d inverse scattering problem and give an analysis of what is proved and what is not proved in this theory

on a structure of the explicit solutions to the davey stewartson equations abstract a general representation of explicit solutions to the davey stewartson ds equations is obtained within the framework of the dressing method the structure of the solutions with nonzero boundary values at infinity is established to coincide with the functional structure of the solutions to equations of the generalized kp hierarchy the exponential and rational type solutions are found both for the ds equations and for their one dimensional reduction the nonlinear schrodinger equation

on the inverse scattering transform of the 2 1 toda equation abstract the method of solution to the 2 1 dimensional toda equation is described in some detail this equation reduces directly to the well know toda lattice in 1 1 dimension and by an appropriate asymptotic reduction to the kadomtsev petviashvili equation in a continuous limit the solution exhibits a number of interesting aspects depending on certain choices of signs of which there are four in the equation for two choices of sign the equation is well posed and linearly stable unstable for the other choices of sign the equation is linearly illposed in these cases we can relate the solution to a boundary value problem and give a formal construction of the solution for one choice of signs in the illposed case an analogue of the sommerfeld radiation condition is developed in order to identify a unique solution in general the method of solution of the toda molecule equation requires an implementation of the dbar technique to cases where the associated eigenfunctions possess both smooth regions of nonholomorphicity and a discontinuity across a curve which in this problem is the unit circle special lump type solutions and solutions depending on suitable arbitrary functions are presented

inverse scattering iii three dimensions continued this paper presents further progress in the solution of the three dimensional inverse scattering problem for the schrodinger equation we prove that if the potential is in a specified class and produces no bound states then the kernel of the generalized marchenko equation defines a compact operator and the equation has a unique solution unless the operator has the eigenvalue 1 a partial characterization of scattering amplitudes associated with underlying local potentials without bound states is given and the potential is constructed without assuming its existence an improved generalization of the marchenko equation is presented for the case with bound states the generalized gel fand levitan equation is critically reviewed

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

sequential subspace reliability method with univariate revolving integration a new computational framework called the sequential subspace reliability method ssrm is presented this method decomposes the multidimensional random space into multiple two dimensional subspaces in this manner ssrm is able to approximate bivariate interaction effects when the reliability estimate contribution is calculated subspace by subspace the final assessment is updated in a progressive manner the iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest in a decision making situation the flexibility of the proposed ssrm to provide iterative updates on reliability estimation becomes especially valuable in dealing with large scale and complex problems under the constraints of limited time and resources to calculate the individual subspace contributions a novel univariate revolving integration uri method is proposed the uri method takes advantage of the axisymmetric nature of a joint pr

inverse scattering iv three dimensions generalized marchenko construction with bound states and generalized gel fand levitan equations this paper represents the final installment in a series on the solution of the inverse scattering problem for the schrodinger equation in three dimensions the potential is constructed from a given scattering amplitude without assuming its existence even in the presence of bound states for exponentially decreasing potentials properties of the jost function and of the regular solution are derived that are sufficient to establish the triangularity of the kernel on which the generalized gel fand levitan gl equation is based other generalized gl equations for nonzero reference potentials and a nonlinear equation are derived and for central potentials they are shown to reduce to the well known radial equations the contents of the series of papers is summarized

davey stewartson type equations in 4 2 and 3 1 possessing soliton solutions an integrable generalisation of a davey stewartson type system of two equations involving two scalar functions in 4 2 i e in four spatial and two temporal dimensions has been recently derived by one of the authors here we first show that there exists a reduction of this system to a single equation involving a scalar function in 4 2 we will refer to this equation as the 4 2 davey stewartson equation we then show that it is possible to reduce this equation to an equation in 3 1 which we will refer to as the 3 1 davey stewartson equation furthermore by employing the so called direct linearising method we compute 1 and 2 soliton solutions for both the 4 2 and the 3 1 davey stewartson equations

new abundant wave solutions of the conformable space time fractional 4 1 dimensional fokas equation in water waves abstract a nonlinear fractional model arising in water waves theory namely the new conformable space time fractional 4 1 dimensional fokas equation is explored via some recently developed approaches namely the functional variable the generalized kudryashov the jacobi elliptic function expansion and the generalized riccati equation mapping methods as a result several abundant distinct types of new exact solutions are constructed with the aid of symbolic computation via mathematica 9 in terms of solitary wave periodic wave jacobi elliptic function and other solutions furthermore some graphical representations numerical simulations have been provided to illuminate the physical behavior of the resultant solutions that may be very important in real world problems and applications with fractional order

nonlocalized modulation of periodic reaction diffusion waves the whitham equation in a companion paper we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized l1 perturbation here we determine time asymptotic behavior under such perturbations showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated whitham averaged equation

some reductions of the self dual yang mills equations to integrable systems in 2 1 dimensions a reduction of the self dual yang mills sdym equations is studied by imposing two space time symmetries and by requiring that the connection one form belongs to a lie algebra of formal matrix valued differential operators in an auxiliary variable in this article the scalar case and the canonical cases for 2 2 matrices are examined in the scalar case it is shown that the field equations can be reduced to the forced burgers equation in the matrix case several well known 2 1 integrable equations are obtained also examined are certain transformation properties between the solutions of some of these 2 1 equations

m lump solutions to a 3 1 dimensional nonlinear evolution equation abstract this paper aims at computing the m lump solutions which decay to a uniform state in all directions for a 3 1 dimensional nonlinear evolution equation these solutions are constructed by taking a long wave limit of the corresponding n soliton solutions obtained by direct methods the dynamic properties of m lump solutions describing multiple collisions of lumps are presented in addition we investigate the interaction between stripe solitons and lumps which is further discussed implying that lumps will be drowned or swallowed by the stripe solitons finally the dynamic properties of interactive wave solutions are graphically depicted by choosing the values of parameters

accuracy improvement of the most probable point based dimension reduction method using the hessian matrix summary this paper proposes a most probable point mpp based dimension reduction method drm using the hessian matrix called hedrm to improve accuracy of reliability analysis in existing mpp based drm methods conventional mpp based drms contain two types of errors 1 error due to eliminating cross terms of a performance function by using the univariate drm 2 error because of dependency of an axis direction after a rotational transformation the proposed method minimizes the aforementioned errors by utilizing the hessian matrix of a performance function by performing an orthogonal transformation using the eigenvectors of the hessian matrix the cross term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the gaussian quadrature points for numerical integration are arranged along the eigenvector directions in this way the error incurred by exiting mpp based drms can be reduced that leads to more accurate probability of failure estimation in addition this paper proposes to allocate the gaussian quadrature points using the magnitude of the eigenvalues of the hessian matrix this allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently copyright 2016 john wiley sons ltd

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

complexification and integrability in multidimensions the complexification of the independent variables of nonlinear integrable evolution partial differential equations pdes in two space dimensions like the celebrated kadomtsev petviashvili and davey stewartson ds equations yields nonlinear integrable equations in genuine 4 2 namely in four real space dimensions x1 x2 y1 y2 and two real time dimensions t1 t2 as opposed to two complex space dimensions and one complex time dimension the associated initial value problem for such equations namely the problem where the dependent variables are specified for all space variables at t1 t2 0 can be solved via a non local d bar formalism here the details of this formalism for the 4 2 ds system are presented furthermore the linearised version of the 3 1 reduction of the 4 2 ds system is discussed the construction of the nonlinear 3 1 reduction remains open in spite of the fact that multi soliton solutions for the 3 1 ds system already exist

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

two types of generalized integrable decompositions and new solitary wave solutions for the modified kadomtsev petviashvili equation with symbolic computation the modified kadomtsev petviashvili mkp equation is shown in this paper to be decomposable into the first two soliton equations of the 2n coupled chen lee liu and kaup newell hierarchies by respectively nonlinearizing two sets of symmetry lax pairs in these two cases the decomposed 1 1 dimensional nonlinear systems both have a couple of different lax representations which means that there are two linear systems associated with the mkp equation under the same constraint between the potential and eigenfunctions for each lax representation of the decomposed 1 1 dimensional nonlinear systems the corresponding darboux transformation is further constructed such that a series of explicit solutions of the mkp equation can be recursively generated with the assistance of symbolic computation in illustration four new families of solitary wave solutions are presented and the relevant stability is analyzed

explicit construction of solutions of the modified kadomtsev petviashvili equation abstract given a solution of the kadomtsev petviashvili equation we explicitly construct a solution of the modified kadomtsev petviashvili equation related to one another by a generalized miura transformation the construction is modeled after a previous treatment of the modified korteweg de vries case as an illustration of our method we derive the soliton solutions of the modified kadomtsev petviashvili equation

integrable nonlinear evolution partial differential equations in 4 2 and 3 1 dimensions the derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s the celebrated korteweg de vries and nonlinear schrodinger equations as well as the kadomtsev petviashvili kp and davey stewartson ds equations are prototypical examples of integrable evolution equations in one and two spatial dimensions respectively do there exist integrable analogs of these equations in three spatial dimensions in what follows i present a positive answer to this question in particular i first present integrable generalizations of the kp and ds equations which are formulated in four spatial dimensions and which have the novelty that they involve complex time i then impose the requirement of real time which implies a reduction to three spatial dimensions i also present a method of solution

new exact traveling wave solutions of the 4 1 dimensional fokas equation abstract in this paper two distinct methods are applied to look for exact traveling wave solutions of the 4 1 dimensional nonlinear fokas equation namely the modified simple equation method msem and the extended simplest equation method esem some new exact traveling wave solutions involving some parameters are obtained the solitary wave solutions can be extracted by assigning special values of these parameters the obtained solutions show the simplicity and efficiency of the used approaches that can be applied for nonlinear equations as well as linear ones

yang mills equations and the inverse scattering transform the authors consider the self dual yang mills system along with its hyperbolic version the author solves the corresponding initial value problem in the second case and a boundary value problem for the self dual yang mills equation

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

linearisable nonlinear partial differential equations in multidimensions we show that the complexification of the independent variables of a given partial differential equation pde provides a straightforward approach for both unifying certain large classes of pdes as well as generating new classes of linearisable pdes in multidimensions for example by complexifying the burgers equation which is a linearisable pde in 1 1 i e an evolution equation in one spatial dimension it is possible to construct a linearisable pde in n 1 i e an evolution pde in n 1 spatial dimensions among these equations in 2 1 are certain linearisable pdes which have been recently introduced in the literature

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

diffusive stability of spatially periodic solutions of the brusselator model applying the lyapunov schmidt reduction approach introduced by mielke and schneider in their analysis of the fourth order scalar swift hohenberg equation we carry out a rigorous small amplitude stability analysis of turing patterns for the canonical second order system of reaction diffusion equations given by the brusselator model our results confirm that stability is accurately predicted in the small amplitude limit by the formal ginzburg landau amplitude equations rigorously validating the standard weakly unstable approximation and the eckhaus criterion

inverse analysis method using mpp based dimension reduction for reliability based design optimization of nonlinear and multi dimensional systems abstract there are two commonly used analytical reliability analysis methods linear approximation first order reliability method form and quadratic approximation second order reliability method sorm of the performance function the reliability analysis using form could be acceptable in accuracy for mildly nonlinear performance functions whereas the reliability analysis using sorm may be necessary for accuracy of nonlinear and multi dimensional performance functions even though the reliability analysis using sorm may be accurate it is not as much used for probability of failure calculation since sorm requires the second order sensitivities moreover the sorm based inverse reliability analysis is rather difficult to develop this paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second order sensitivities for reliability based design optimization rbdo of nonlinear and multi dimensional systems for the inverse reliability analysis the most probable point mpp based dimension reduction method drm is developed since the form based reliability index is inaccurate for the mpp search of the nonlinear performance function a three step computational procedure is proposed to improve accuracy of the inverse reliability analysis probability of failure calculation using constraint shift reliability index update and mpp update using the three steps a new drm based mpp is obtained which estimates the probability of failure of the performance function more accurately than form and more efficiently than sorm the drm based mpp is then used for the next design iteration of rbdo to obtain an accurate optimum design even for nonlinear and or multi dimensional system since the drm based rbdo requires more function evaluations the enriched performance measure approach pma with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations

diffusive mixing of periodic wave trains in reaction diffusion systems abstract we consider reaction diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u 0 k x t k that are parameterized by the wave number k we prove stable diffusive mixing of the asymptotic states u 0 k x k as x with different phases at infinity for solutions that initially converge to these states as x the proof is based on bloch wave analysis renormalization theory and a rigorous decomposition of the perturbations of these wave solutions into a phase mode which shows diffusive behavior and an exponentially damped remainder depending on the dispersion relation the asymptotic states mix linearly with a gaussian profile at lowest order or with a nonsymmetric non gaussian profile given by burgers equation which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation

considerations on conserved quantities and boundary conditions of the 2 1 dimensional nonlinear schrodinger equation abstract in this study we consider a natural integrable generalization of the defocusing cubic nonlinear schrodinger equation to two dimensions and we classify the admissible boundary conditions in particular we determine whether the classical physical observables are conserved mass momentum and hamiltonian we find that this is the case when a certain integral the mass constraint vanishes the vanishing of the mass constraint and thus the existence of conserved quantities is contingent on the boundary conditions adopted in particular under decaying boundary conditions the hamiltonian is not necessarily conserved

constructing lump like solutions of the hirota miwa equation we construct special solutions of the hirota miwa equation for which the function is a polynomial in the independent variables three different methods are presented direct construction obtained also as a limit of the soliton solutions and the derivation of the solutions in two different determinant forms namely grammian and casorati introducing the appropriate ansatz we write the hirota miwa equation in a nonlinear form for a single variable in terms of the latter the solutions obtained are rational and are reminiscent of the lump solutions for the continuous analogue of the hirota miwa equation namely the kp equation

the d bar method inversion of certain integrals and integrability in 4 2 and 3 1 dimensions we first review a method for deriving linear and nonlinear transform pairs which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d bar problem then we present two applications of this method a we derive a certain linear transform pair in one dimension which appears in the characterization of the dirichlet to neumann map of the laplace equation in the interior of a convex two dimensional curvilinear domain b we derive a nonlinear fourier transform pair in four dimensions which can be used for the solution of the cauchy problem of an integrable generalization of the kadomtsev petviashvilli equation in 4 2 i e in four spatial and two temporal dimensions the question of reducing this equation form 4 2 to 3 1 dimensions is also discussed

towards nonlinear stability of sources via a modified burgers equation coherent structures are solutions to reaction diffusion systems that are time periodic in an appropriate moving frame and spatially asymptotic at x to spatially periodic travelling waves this paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system determining their nonlinear stability properties is challenging as localized perturbations may lead to a non localized response even on the linear level due to the outward transport using a burgers type equation as a model problem that captures some of the essential features of sources we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context

on an algorithmic construction of lump solutions in a 2 1 integrable equation the singular manifold method is used to generate lump solutions of a generalized integrable nonlinear schrodinger equation in 2 1 dimensions we present several essentially different types of lump solutions the connection between this method and the ablowitz villarroel scheme is also analysed

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

rational solutions of the kp hierarchy and the dynamics of their poles ii construction of the degenerate polynomial solutions a general approach to constructing the polynomial solutions satisfying various reductions of the kadomtsev petviashvili kp hierarchy is described within this approach the reductions of the kp hierarchy are equivalent to certain differential equations imposed on the function of the hierarchy in particular the l reduction and the k constraint as well as their generalized counterparts are considered a general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the kdv and gardner equations the boussinesq and classical boussinesq systems the nls and yajima oikawa equations it is shown that the degenerate rational solutions of the kp hierarchy are related to stationary manifolds of the calogero moser cm hierarchy of dynamical systems the scattering dynamics of interacting particles in the cm systems may become complicated due to an anomalously slow fractional power rate of the particle motion along the stationary manifolds

toda lattice mass transport in lagrangian mechanics and in a two dimensional system abstract this paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the toda lattices particular attention is paid to the broken isotropy in the kdv and burgers equations the flow variable representation is established from the lagrangian mechanics for hydrodynamic mass transport based on the inverse scattering transform the gel fand levitan marchenko glm equation is formulated from the kdv equation expressed by the flow variable representation we found that a kernel of the glm equation is given by the concentration variable q x t a lagrangian is formulated for the kdv equation in state space q x t k x t next an extension of the flow variable representation is sought in a two dimensional system the lhs of the kadomtsev petviashvili kp equation takes the same form as in the second formalism of the kdv equation by setting up the flow variable representation of the kp equation the burgers equation in two dimensions is formulated these results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations these results provide physical insight into various consequences of the generalized form of the kawasaki ohta equation from the viewpoint of mass transport

exotic coherent structures in the 2 1 dimensional long dispersive wave equation in this paper we investigate the integrability aspects of the 2 1 dimensional coupled long dispersive wave 2ldw equation introduced recently by chakravarty kent and newman and establish its painleve p property we then deduce its bilinear form from the p analysis and use it to construct wave type solutions for the field variables we then identify line solitons for the composite field variable qr which eventually helps to bring out the peculiar localization behavior of the system by generating localized structures dromions for the composite field from out of only one ghost soliton driving the boundary we have then extended this analysis to multidromion solutions

interaction of lumps with a line soliton for the dsii equation abstract exact solutions of the focussing davey stewartson ii equation are presented which describe the interaction of n lumps with a line soliton these solutions are constructed by analysing the inverse spectral problem of the associated lax pair the dynamical properties of these solutions are also discussed

pointwise nonlinear stability of nonlocalized modulated periodic reaction diffusion waves abstract in this paper extending previous results of 2 we obtain pointwise nonlinear stability of periodic traveling reaction diffusion waves assuming spectral linearized stability under nonlocalized perturbations more precisely we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically

on the discrete spectrum of the nonstationary schrodinger equation and multipole lumps of the kadomtsev petviashvili i equation the discrete spectrum of the nonstationary schrodinger equation and localized solutions of the kadomtsev petviashvili i kpi equation are studied via the inverse scattering transform it is shown that there exist infinitely many real and rationally decaying potentials which correspond to a discrete spectrum whose related eigenfunctions have multiple poles in the spectral parameter an index or winding number is asssociated with each of these solutions the resulting localized solutions of kpi behave as collection of individual humps with nonuniform dynamics

on the solution to the inverse problem for the toda chain the elliptic version of a three dimensional nonlinear integrable equation the toda chain equation is studied by means of the inverse scattering method several classes of decaying solutions corresponding to both the continuous and discrete spectrum of the associated spectral problem are obtained conditions that guarantee reality and analyticity along with relevant estimates are also given finally a class of solutions with rotational symmetry is constructed

the time dependent schrodinger equation of dimension k 1 explicit and rational solutions via gbdt and multinodes a version of the binary darboux transformation is constructed for a non stationary schrodinger equation of dimension k 1 where k 1 is the number of space variables this is an iterated generalized backlund darboux transformation version new families of non singular and rational potentials and solutions are obtained some results are also new for the case that k 1 a certain generalization of a colligation introduced by m s livsic and a generalization of the s node introduced by l a sakhnovich have been successfully used in our construction

pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws abstract in the previous paper 9 we showed time asymptotic behavior with detailed decaying rates of perturbations of periodic traveling reaction diffusion waves under small initial perturbations with a gaussian rate and an algebraic rate here we establish pointwise nonlinear stability up to an appropriate modulation of periodic traveling waves of systems of viscous conservation laws under small algebraic decaying initial data similar to the reaction diffusion equations by using bloch decomposition we start with pointwise bounds on the green function of the linearized operator about underlying solutions

the novikov veselov equation and the inverse scattering method part i analysis abstract the novikov veselov nv equation is a 2 1 dimensional nonlinear evolution equation that generalizes the 1 1 dimensional korteweg de vries kdv equation the solution of the nv equation using the inverse scattering method has been discussed in the literature but only formally or with smallness assumptions in the case of nonzero energy because of the possibility of exceptional points or singularities in the scattering data in this work absence of exceptional points is proved at zero energy for evolutions with compactly supported smooth and rotationally symmetric initial data of the conductivity type q 0 1 2 1 2 with a strictly positive function the inverse scattering evolution is shown to be well defined real valued and preserving conductivity type there is no smallness assumption on the initial data

the structure of the rational solutions to the boussinesq equation rational solutions of the one dimensional boussinesq equation both with zero and nonzero boundary conditions at the infinity are obtained by reducing the known solutions of the kadomtsev petviashvili equation the structure of the found solutions generalizes a family of rational solutions to the korteweg de vries equation to the case of two wave processes

combined partial bar and riemann hilbert inverse methods for integrable non linear evolution equations in 2 1 dimensions the authors give a natural combination of partial bar and riemann hilbert problem inverse methods for an n th order scalar spectral problem which solves a number of integrable non linear evolution equations in two space and one time 2 1 dimensions the theory embraces the two kadomtsev petviashvili equations 1970 and their lump solutions

polynomial chaos expansion with latin hypercube sampling for estimating response variability a computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented to deal with the random nature of input parameters of structural models several efficient probabilistic methods are investigated specifically the polynomial chaos expansion with latin hypercube sampling is used to represent the response of an uncertain system latin hypercube sampling is employed for evaluating the generalized fourier coefficients of the polynomial chaos expansion because the key challenge in uncertainty analysis is to find the most significant components that drive response variability analysis of variance is employed to find the significant parameters of the approximation model several analytical examples and a large finite element model of a joined wing are used to verify the effectiveness of this procedure

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

new 2 1 dimensional nonisospectral toda lattice hierarchy we give a new 2 1 dimensional nonisospectral generalization of the toda lattice hierarchy reductions yield a variety of new integrable hierarchies along with their underlying linear problems including new 1 1 dimensional differential delay hierarchies nonisospectral and isospectral new ordinary differential delay hierarchies and new discrete painleve hierarchies we also show that a reduction in components yields our previously obtained 2 1 dimensional nonisospectral volterra lattice hierarchy

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

nondegeneracy morse index and orbital stability of the kp i lump solution using backlund transformation we analyze the spectral property of the kp i lump solution it is proved that the lump is nondegenerate and its morse index equals one as a consequence we show that it is orbitally stable

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

source generation of the davey stewartson equation the source generation procedure sgp proposed by hu and wang inverse probl 22 1903 2006 provides a new way to systematically generate so called soliton equations with self consistent sources in this paper we apply this sgp to a davey stewartson ds equation based on the hirota bilinear form producing a system of equations which is called the ds equation with self consistent sources dsescs meanwhile we obtain the gramm type determinant solutions to the dsescs since the ds equation is a 2 1 dimensional integrable generalization of the nonlinear schrodinger nls equation the dsescs may be viewed as a 2 1 dimensional integrable generalization of the nonlinear schrodinger equation with self consistent sources these results indicate the commutativity of source generation procedure and 2 1 dimensional integrable generalizations for the nls equation

on the integrability of the nonlinear schrodinger equation with randomly dependent linear potential we show that the cauchy problem for the nonlinear schrodinger equation driven by a term of the form where w t is a brownian motion can be integrated by means of the inverse scattering transform the solution is written in terms of w t and the integral of powers j 1 2 a complete set of integrals of the motion is determined we relate the probability density of the solution to the green s function of a linear partial differential operator with quadratic coefficients

on generalisation of the backlund calogero transformations for integrable equations it is shown that the consideration of dressing gauge transformations non local on all spatial variables and spectral parameters allows one to extend the class of general backlund calogero transformations for the kadomtsev petviashvili equation

diffusive stability against nonlocalized perturbations of planar wave trains in reaction diffusion systems abstract planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction in this paper we investigate the stability of planar wave trains in reaction diffusion systems we establish nonlinear diffusive stability against perturbations that are bounded along a line in r 2 and decay exponentially in the distance from this line our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation we also consider perturbations that are fully localized and establish nonlinear stability with better decay rates suggesting a trade off between spatial localization of perturbations and temporal decay rate our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations the nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme instead we track the perturbed solution in two different coordinate systems

parameter limit method and its application in the 4 1 dimensional fokas equation abstract a new hirota s bilinear form of the 4 1 dimensional fokas equation is obtained by an appropriate wave transformation applying the parameter limit method introduced in this paper we obtain two homoclinic solutions and a rogue wave solution of 4 1 dimensional fokas equation besides we also discuss the interaction between rogue wave solution and different forms of stripe solitons one stripe soliton and double stripe solitons and investigate the annihilation and emergence of rogue waves finally the parametric reasons of soliton fusion and fission are discussed furthermore the fusion and fission phenomena of soliton are simulated by three dimensional plots

pointwise asymptotic behavior of modulated periodic reaction diffusion waves abstract by working with the periodic resolvent kernel and the bloch decomposition we establish pointwise bounds for the green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations with our linearized estimates together with a nonlinear iteration scheme developed by johnson zumbrun we obtain l p behavior p 1 of a nonlinear solution to a perturbation equation of a reaction diffusion equation with respect to initial data in l 1 h 2 recovering and slightly sharpening results obtained by schneider using weighted energy and renormalization techniques we obtain also pointwise nonlinear estimates with respect to two different initial perturbations u 0 e 0 e x 2 m u 0 h 2 e 0 and u 0 e 0 1 x r r 2 u 0 h 2 e 0 respectively e 0 0 sufficiently small and m 1 sufficiently large showing that behavior is that of a heat kernel these pointwise bounds have not been obtained elsewhere and do not appear to be accessible by previous techniques

painleve analysis of new higher dimensional soliton equation in this note we prove that the recently proposed new higher dimensional nonlinear partial differential equation admits the painleve property we briefly discuss the integrability properties of the equation

the kadomtsev petviashvili equation as a source of integrable model equations a new integrable and nonlinear partial differential equation pde in 2 1 dimensions is obtained by an asymptotically exact reduction method based on fourier expansion and spatiotemporal rescaling from the kadomtsev petviashvili equation the integrability property is explicitly demonstrated by exhibiting the corresponding lax pair that is obtained by applying the reduction technique to the lax pair of the kadomtsev petviashvili equation this model equation is likely to be of applicative relevance because it may be considered a consistent approximation of a large class of nonlinear evolution pdes

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

painleve analysis of the damped driven nonlinear schrodinger equation on applique les tests de painleve a l equation iu t u xx 2 u 2 u a x t u b x t ou a et b sont des fonctions analytiques de x et t pour etudier sous quelles conditions l equation peut etre completement integrable

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations