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

new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

regularization and error estimate for an initial inverse nonlocal diffusion problem abstract we study the backward problem of determining the initial condition for a system of parabolic diffusion equations which is severely ill posed in the sense of hadamard to stabilize the solution we develop the quasi reversibility qr and fourier truncation methods to construct the regularized solutions we also investigate the error estimates and convergence rates between the regularized solutions and the true solution in l 2 and h 1 norm a numerical scheme is presented

decay of the navier stokes poisson equations we establish the time decay rates of the solution to the cauchy problem for the compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates as a corollary we also obtain the usual lp l2 1 p 2 type of the optimal decay rates compared to the compressible navier stokes system and the compressible irrotational euler poisson system our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

remarks on the painleve analysis for a 2 1 dimensional generalization of nonlinear schrodinger type equations choudhury j math phys 44 5733 2003 applied truncated painleve expansions to derive lax pairs darboux transformations and various soliton solutions to some 2 1 dimensional generalizations of nonlinear schrodinger type equations and the method he used was based on the techniques of painleve analysis however there are some errors in the procedures of the painleve analysis for one equation which was investigated in detail

the cauchy problem of coupled elliptic sine gordon equations with noise analysis of a general kernel based regularization and reliable tools of computing abstract developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering in this paper we explore the regularization method for the coupled elliptic sine gordon equations along with cauchy data the system of equations originates from the static case of the coupled hyperbolic sine gordon equations modeling the coupled josephson junctions in superconductivity and so far it addresses the josephson junctions in general the cauchy problem is not well posed and herein the hadamard instability occurs drastically generalizing the kernel based regularization method we propose a stable approximate solution confirmed by the error estimate this solution strongly converges to the exact solution in l 2 norm the main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation we employ the proposed techniques that successfully approximated the highly oscillatory integral and apply the picard like iteration to organize an efficient and reliable tool of computations the results are viewed as the improvement as well as the generalization of many previous works the paper is also accompanied by a numerical example that demonstrates the potential of this idea

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

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

periodic fixed points of backlund transformations and the korteweg de vries equation a new method for studying integrable systems based on the periodic fixed points of backlund transformations bt s is presented normally the bt maps an old solution into a new solution and requires a known seed solution to get started besides this limitation it can also be difficult to qualitatively classify the result of applying the bt several times to a known solution by studying the periodic fixed points of the bt regarded as a nonlinear map in a function space integrable systems of equations of finite degree equal to the order of the fixed point and a method for the systematic classification of the solutions of the original system are obtained

extension of the module of invertible transformations classification of integrable systems we demonstrate that for the systems of equations which are invariant under a point group or possess conservation laws of the zeroth or first order a nontrivial extension of the module of invertible transformations is possible that simplifies greatly a classification of the integrable systems of equations here we present an exhaustive list and a classification of the second order systems of the formu t u xx f u v u x v x v t v xx g u v u x v x which possess the conservation laws of higher order the reduction group approach allows us to define the lax type representations for some new equations of our list

generalized dromion solutions of the 2 1 dimensional kdv equation the rich dromion structures for a 2 1 dimensional kdv equation are revealed the dromions in a high dimensional integrable model may have a free shape in one or more directions multi dromion solutions can be driven by perpendicular line non perpendicular line and curved line ghost solitons

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

a nonlinear parabolic equation backward in time regularization with new error estimates abstract consider a nonlinear backward parabolic problem in the form u t a u t f t u t u t g where a is a positive self adjoint unbounded operator based on the fundamental solution to the parabolic equation we propose to solve this problem by the fourier truncated method which generates a well posed integral equation then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven our regularizing scheme can be considered a new regularization with the advantage of a relatively small amount of computation compared with the quasi reversibility or quasi boundary value regularizations error estimates for this method are provided together with a selection rule for the regularization parameter these errors show that our method works effectively

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

algorithmic method for deriving lax pairs from the invariant painleve analysis of nonlinear partial differential equations given a partial differential equation its painleve analysis will first be performed with a built in invariance under the homographic group acting on the singular manifold function then assuming an order for the underlying lax pair a multicomponent pseudopotential of projective riccati type the components of which are homographically invariant is introduced if the equation admits a classical darboux transformation a very small set of determining equations whose solution yields the lax pair will be generated in the basis of the pseudopotential this new method will be applied to find the yet unpublished lax pair of the scalar hirota satsuma equation

on the structure and properties of the singularity manifold equations of the kp hierarchy this paper is concerned with the painleve expansion and the singularity manifold sm equation of the kadomtsev petviashvili kp equation several aspects of the interrelation between the sm equation and the kp auxiliary linear system are studied it is shown that the simultaneous painleve expansion for the kp potential u and the kp eigenfunction can be treated as a backlund gauge transformation two methods for the derivation of the sm equation based on this treatment are proposed and their equivalence is proved the interrelation between the sm equation and the vertical hierarchy of the kp eigenfunction equations is discussed the explanation of the coincidence of the kp eigenfunction equation of the second level and the kp sm equation is given compact forms of the hierarchy of sm equations of the kp hierarchy are presented the connection between the kp singularity manifold function and the kp eigenfunctions and the adjoint kp eigenfunctions is derived the bilinear bilocal description of th

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

decay of the non isentropic navier stokes poisson equations abstract we establish the time decay rates of the solution to the cauchy problem for the non isentropic compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained as a corollary we also obtain the usual l p l 2 1 p 2 type of the optimal decay rates the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates we use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

global existence for compressible navier stokes poisson equations in three and higher dimensions the compressible navier stokes poisson system is concerned in the present paper and the global existence and uniqueness of the strong solution is shown in the framework of hybrid besov spaces in three and higher dimensions

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

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

periodic fixed points of backlund transformations the periodic fixed points of the backlund transformations are finite dimensional invariant manifolds for the flow of the system the dynamics occur as commuting hamiltonian flows on this finite dimensional manifold we examine the flow of the kdv periodic fixed points in the neighborhood of steady states and reductions these are analogous to a flow in the neighborhood of a sequence of heteroclinic points

a stability estimate on the regularized solution of the backward heat equation inverse problems where one looks for causes for observed or desired effects are usually ill posed i e their solution is not unique or unstable with respect to data perturbations we derive a stability estimate for the regularized solution of the backward heat equation which is needed for stable solution of this ill posed problem for this purpose we construct a regularizer for a certain minimization problem and in conclusion the stability of this minimization procedure

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

on a class of problems involving a nonlocal operator in this paper we prove a result on existence of positive solution for a nonlocal elliptic problem by using a result on fixed point index theory and we establish another existence result which is proved by sub and supersolution and a comparison principle moreover we prove the existence uniqueness and asymptotic behaviour of the solutions for the evolution case

integrability characteristics of two dimensional generalizations of nls type equations a recent procedure based on truncated painleve expansions is used to derive lax pairs darboux transformations and various soliton solutions for integrable 2 1 generalizations of nls type equations in particular diverse classes of solutions are found analogous to the dromion instanton lump and ring soliton solutions derived recently for 2 1 korteweg de vries type equations the nizhnik novikov veselov equation and the 2 1 broer kaup system

long time behavior of a non autonomous parabolic equation with nonlocal diffusion and sublinear terms abstract this paper is devoted to study the asymptotic behavior of a time dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth namely we extend some previous results from the literature obtaining existence uniqueness and continuity results analyzing the stationary problem and decay of the solutions of the evolutionary problem and finally under more general assumptions ensuring the existence of pullback attractors for the associated dynamical system in both l 2 and h 1 norms relationships among these objects are established using regularizing properties of the equation

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure