asymptotic behavior of the solution to the system for a viscous reactive gas abstract in this paper we investigate the asymptotic behavior of the generalized solution to the compressible navier stokes equations which describes a thermal explosion in a viscous reactive perfect gas confined between two infinite parallel plates in combustion theory

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

solitons in a nonlinear model medium the periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short wave perturbations in a relaxing medium are studied solutions expressed by a multiple valued function are shown to exist a method for determining the nonlinear interaction between solitary waves is suggested an example of a collision of solitons is given

oscillation and global attractivity in a delay periodic host macroparasite model abstract in this paper we shall consider the nonlinear delay host macroparasite model with periodic coefficients with common period by means of a continuation theorem in coincidence degree theory we establish a sufficient condition for the existence of a positive periodic solution m t with strictly positive components also we establish some sufficient conditions for oscillation of all positive solutions about the positive periodic solution m t and establish a sufficient condition for the global attractivity of m t

global existence of large solutions to initial boundary value problems for a viscous heat conducting one dimensional real gas abstract the existence of global classical solutions to initial boundary value problems in the dynamics of a one dimensional viscous heat conducting gas is established the nonlinear dissipative effects turn out to be sufficiently strong to prevent the development of singularities

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

oscillation analysis of numerical solutions for delay differential equations with real coefficients abstract the paper is mainly concerned with the oscillations of numerical solutions by the linear methods for a kind of linear delay differential equations with positive and negative coefficients some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions numerical experiments are given to demonstrate our results

darboux transformation for the modified veselov novikov equation a darboux transformation is constructed for the modified veselov novikov equation by means of the darboux transformation two families of explicit solutions of this equation are given

global entropy solutions to exothermically reacting compressible euler equations abstract the global existence of entropy solutions is established for the compressible euler equations for one dimensional or plane wave flow of an ideal gas which undergoes a one step exothermic chemical reaction under arrhenius type kinetics we assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas the heat released by the reaction causes the spatial total variation of the solution to increase however the increase in total variation is proved to be bounded in t 0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity

nonlinear schrodinger type equations from multiscale reduction of pdes ii necessary conditions of integrability for real pdes the multiscale method has been applied in j math phys 41 6399 2000 to evolution pdes to obtain various nonlinear schrodinger type equations here these results are exploited to derive necessary conditions of integrability for nonlinear real evolution pdes in 1 1 dimensions these conditions are given by explicit formulas and several examples are detailed

oscillation in first order neutral differential equations with integrally small coefficients abstract consider the neutral differential equation with positive and negative coefficients d dt x t r t x t r p t x t q t x t 0 where p q r c t0 r r 0 and r we obtain some new sufficient conditions for the oscillation of solutions of the above equation without the restriction t 0 p s q s ds

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

dromion can be remote controlled the one dromion solution of the davey stewartson 1 equations with driving boundaries is analysed numerically it is shown that the dromion can follow the motion of the crosspoint of the mean flows even if it is not of exact solutions moreover it is found that the localized structure keeps its shape when it is exposed to the forced motion of the mean flows in the case where dromion runs on a circle path the decay ratio of the localized structure versus the radius is given several types of movement of mean flows are also studied it is expected that dromion can be controlled arbitrarily by driving mean flows at the boundaries

universal and integrable nonlinear evolution systems of equations in 2 1 dimensions integrable systems of nonlinear partial differential equations pdes are obtained from integrable equations in 2 1 dimensions by means of a reduction method of broad applicability based on fourier expansion and spatio temporal rescalings which is asymptotically exact in the limit of weak nonlinearity the integrability by the spectral transform is explicitly demonstrated because the corresponding lax pairs have been derived applying the same reduction method to the lax pair of the initial equation these systems of nonlinear pdes are likely to be of applicative relevance and have a universal character inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part

the investigation into new integrable systems of equations in 2 1 dimensions a new integrable class of systems of nonlinear partial differential equations npdes in 2 1 dimensions is derived from the matrix nizhnik novikov veselov nvv equation by means of an asymptotically exact nonlinear reduction method based on fourier expansion and spatio temporal rescaling the integrability by the inverse scattering method is explicitly demonstrated by applying the reduction technique also to the lax pair of the starting matrix equation and thereby obtaining the lax pair for the new class of systems of equations a reduction to a system of two interacting complex fields is briefly described

new oscillation criteria for linear delay differential equations abstract we obtain new sufficient conditions for the oscillation of all solutions of the first order linear delay differential equation x t p t x t 0 with p t 0 t t our results improve the known results in the literature which include those obtained recently in papers by elbert and stavroulakis li and yu et al

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

localized solutions with chaotic and fractal behaviours in a 2 1 dimensional dispersive long wave system it is known that in high dimensions there exist abundant localized excitations such as dromions lumps ring soliton solutions and so on in this paper the possible chaotic and fractal localized structures are revealed for the 2 1 dimensional dispersive long wave equation the chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower dimensional chaotic and fractal patterns

new nonoscillation criteria for delay differential equations abstract this paper is concerned with the nonoscillation of first order delay differential equations of the form x t p t x t 0 t t 0 where p c t 0 r r 0 t is nondecreasing t t for t t 0 and lim t t new nonoscillation conditions are obtained these conditions concern the case when the well known nonoscillation condition t t p s ds 1 e is not satisfied

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

a multidimensional model for the combustion of compressible fluids we consider a multidimensional model for the combustion of compressible reacting fluids the flow is governed by the navier stokes in eulerian coordinates and the chemical reaction is irreversible and is governed by the arrhenius kinetics the existence of globally defined weak solutions is established by using weak convergence methods compactness and interpolation arguments in the spirit of feireisl 16 and p l lions 24

generalized symmetries and w infinity algebras in 3 dimensional toda field theory after establishing a formal theory for getting solutions of one type of high dimensional partial differential equation two sets of generalized symmetries of the 3d toda theory which arises from a particular reduction of the 4d self dual gravity equation are obtained concretely by a simple formula each set of symmetries constitutes a generalized w algebra which contains three types of the usual w algebras as special cases some open questions are discussed

global solutions of the navier stokes equations for viscous compressible flows the compressible navier stokes equations for viscous flows with general large continuous initial data as well as with large discontinuous initial data are studied both a homogeneous free boundary problem with zero outer pressure and a fixed boundary problem are considered for the large initial data in h1 the existence uniqueness and regularity of global solutions in h1 for real viscous flows are established and it is showed that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time for the large discontinuous data the global existence of large weak solutions for the perfect gases is also established using a different argument and it is indicated that the solutions do not develop vacuum or concentration although the solutions have large discontinuity for the free boundary problem the interface separating the flows from the zero outer pressure expands at a finite speed

nonlinear schrodinger type equations from multiscale reduction of pdes i systematic derivation in this article we begin a systematic investigation via multiscale expansions of nonlinear evolution pdes partial differential equations in this first article we restrict consideration to a single autonomous but otherwise generic pde in 1 1 variables space time of first order in time whose linear part is dispersive and to solutions dominated by a single plane wave satisfying the linear part of the pde the expansion parameter is an assumedly small coefficient multiplying this plane wave the main indeed asymptotically exact effect of the weak nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics which is generally described in appropriately defined coarse grained time and space variables by evolution equations of nonlinear schrodinger type a systematic analysis of such equations is presented corresponding to various assumptions on the resonances occurring for the first few harmonics

2 1 dimensional m n component akns system painleve integrability infinitely many symmetries similarity reductions and exact solutions the 2 1 dimensional m n component akns system that is derived from the inner parameter dependent symmetry constraint of the kp equation is studied in detail first the painleve integrability of the model is proved by using the standard wtc and kruskal approach using the formal series symmetry approach the generalized kmv symmetry algebra and the related symmetry group are found the two dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized kmv symmetry algebra and the direct method abundant localized coherent structures are revealed by the variable separation approach some special types of the localized excitations like the multiple solitoffs dromions lumps ring solitons breathers and instantons are plotted also

global solutions of real compressible reactive gas with density dependent viscosity and self gravitation for higher order kinetics abstract a mathematical model for viscous real compressible reactive fluid flows is considered the existence of global solutions for the free boundary problem with species diffusion in dynamic combustion is established when the viscosity depends on the density i e a 0 1 2 where a is a generic positive constant furthermore the equations of state depend nonlinearly on density and temperature unlike the case of perfect gases or radiative flows in addition the shock wave turbulence vacuum mass concentration or extremely hot spot will not be developed in any finite time if the initial data do not contain vacuum

oscillation and global attractivity in hematopoiesis model with periodic coefficients in this paper we shall consider the nonlinear delay differential equation p t email protected t p m email protected 1 p n email protected email protected t p t where k is a positive integer b t and c t are positive periodic functions of period w in the nondelay case we shall show that has a unique positive periodic solution email protected t and we will study the global attractivity of email protected t in the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of about email protected t and establish some sufficient conditions for the global attractivity of email protected t our results in this paper extend as well as improve the results in the autonomous case

on temporal asymptotics for the pth power viscous reactive gas abstract in this paper we investigate the long time behaviour of solutions to the system governing a heat conductive viscous reactive pth power gas confined between two parallel plates for initial boundary value problems with the end points held at a prescribed temperature or insulated we prove the global existence of physically relevant solutions and establish their rate of convergence to equilibria for generic initial data the estimates for different boundary conditions are presented in a unified manner

global smooth solutions of the compressible navier stokes equations with density dependent viscosity abstract in this paper we study a free boundary problem for the viscous compressible heat conducting one dimensional real fluids more precisely the viscosity is assumed to be a power function of density i e where denotes the density of fluids and is a positive constant in addition the equations of state include and are more general than perfect flows which only depend linearly on temperature the global existence uniqueness of smooth solutions is established with 0 1 2 for general large initial data which improves the previous results moreover it is also shown that the solutions will not develop vacuum mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth and do not contain vacuum

group theoretical analysis of dispersive long wave equations in two space dimensions abstract the symmetry algebra of an integrable dispersive long wave equation in two space dimensions is shown to be infinite dimensional and to have a kac moody virasoro structure the corresponding symmetry group is used to generate a large number of new solutions in particular solitons kinks and periodic waves these waves have wave crests of quite general shapes the results are compared to those for a nonintegrable two space dimensional dispersive long wave equation for which the symmetry group is finite dimensional and wave crests are always straight lines

a new integrable davey stewartson type equation a new integrable nonlinear partial differential equation pde in 2 1 dimensions is derived starting from the konopelchenko dubrovsky equation we use an asymptotically exact reduction method based on fourier expansion and spatio temporal rescaling and obtain a new integrable davey stewartson type equation in order to demonstrate the integrability of the new equation by the inverse scattering method we apply the reduction technique to the lax pair of the konopelchenko dubrovsky equation and find the corresponding lax pair of the new equation the new equation reduces to the davey stewartson or the nonlinear schrodinger equation by appropriate limits

nonlinear evolution equations admitting multilinear variable separable solutions based on the hirota s bilinear transformation method it is shown that a family of nonlinear partial differential equations admit multilinear variable separation solutions this extends the class of nonlinear partial differential equations which admit the multilinear variable separation solutions

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data we prove the global existence of solutions of the navier stokes equations describing the dynamic combustion of a compressible exothermically reacting fluid and we study the large time behavior of solutions giving necessary and sufficient conditions for complete combustion in certain cases the adiabatic constants and specific heats of the burned product and unburned reactant fluids may differ and the initial data may be large and discontinuous

on the large time behavior of 1d radiative and reactive viscous flows for higher order kinetics abstract we consider the system of quasilinear equations describing 1d radiative and reactive viscous flows with arbitrarily large data the large time behavior of solutions in the case of first order kinetics has been recently studied in this paper we present new results concerning the case of higher order kinetics for fairly general kinetics law unbounded with respect to density and temperature and dealing with the ignition phenomenon including l 2 and h 1 stabilization rate bounds of power type the power exponents of bounds improve essentially those known for related problems and are partially proved to be sharp an effect of faster equaling of values in space for the concentration of unburned gas is also found finally we show how our results are modified in the case of reaction without diffusion

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

soliton solutions for two nonlinear partial differential equations using a darboux transformation of the lax pairs two darboux transformations of the 1 1 dimensional wu zhang wz equation and the two component camassa holm 2ch system with the reciprocal transformation are obtained one loop and two loop soliton solutions and multisoliton like solutions of the 2ch system are obtained by using the darboux transformations and selecting different seed solutions of the corresponding equations the bidirectional soliton solutions of the 1 1 dimensional wz equation are also obtained the interactions of two soliton head on and overtaking collisions for the wz equation and the evolution of the two soliton like solutions for the 2ch system are studied

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

energy exchange of interacting coherent structures in multidimensions abstract the formalism recently developed by fokas and the author for the solution of the initial boundary value problem for the davey stewartson equation is used to study the time evolution of the multidromion solution showing in particular that interacting coherent structures in multidimensions not only exhibit a two dimensional phase shift but also change amplitude and exchange energy several examples of physically relevant boundary conditions which generate other kinds of explicit finite energy solutions are also presented

global existence and mann iterative algorithms of positive solutions for first order nonlinear neutral delay differential equations abstract this paper deals with the first order nonlinear neutral delay differential equation d dt x t p t x t f t x 1 t x 2 t x n t 0 t t 0 where 0 p c t 0 r f c t 0 r n r and l c t 0 r with limt l t for l 1 2 n by using the banach fixed point theorem we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p construct some mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions seven examples are presented to illuminate the results obtained in this paper

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

a class of integrable davey stewartson type systems a new integrable davey stewartson type class of systems of partial differential equations in 2 1 dimensions is derived from a previously known integrable equation by means of an asymptotically exact nonlinear reduction method based on fourier expansion and spatio temporal rescaling the integrability by the inverse scattering method is explicitly demonstrated because the reduction technique is applied to the lax pair of the starting equation and the corresponding lax pair of the class of systems of equations is found the particular characteristics of the reduction method imply that the new systems are likely to be of applicative relevance

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis