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

on generalized painleve chains abstract painleve chains are recursively defined sequences of dominant truncations of polynomial partial differential equations in this paper a new kind of painleve chain is defined which can lead to new branches of the painleve expansions associated to the dominant truncations applying methods of hlavaty a recursion relation for the resonances of the members of these chains is deduced a special case is considered in detail

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

some results on the caudrey dodd gibbon kotera sawada equation in this paper a hierarchy of bilinear caudrey dodd gibbon kotera sawada equations with a unified structure is proposed a nonlinear superposition formula for the cdgks equation is proved under certain conditions a backlund transformation for a higher order cdgks equation is presented

singularity structure analysis and hirota s bilinearisation of the davey stewartson equation the singularity structure aspects of the davey stewartson equation is investigated and it is shown that the system possesses the generalised painleve property in the sense of weiss tabor and carnevale 1983 further the associated backlund transformation in constructed and hirota s bilinearisation is obtained straightforwardly from the painleve analysis

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

solving linear systems on a vector computer abstract this paper gives a classification for the triangular factorization of square matrices these factorizations are used for solving linear systems efficient algorithms for vector computers are presented on basis of criteria for optimal algorithms moreover the gauss jordan elimination algorithm in a version which admits efficient implementation on a vector computer is described comparative experiments in fortran 77 with fortran 200 extensions for the cyber 205 are reported

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

the hirota s direct method and the tanh coth method for multiple soliton solutions of the sawada kotera ito seventh order equation abstract in this work the sawada kotera ito seventh order equation is studied the tanh coth method is applied to obtain soliton solution of this equation the hirota s direct method combined with the simplified hereman s method are applied to determine the n soliton solutions for this equation the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

soliton structure versus singularity analysis third order completely intergrable nonlinear differential equations in 1 1 dimensions abstract backlund transformations between all known completely integrable third order differential equations in 1 1 dimensions are established and the corresponding transformations formulas for their hereditary operators and hamiltonian formulations are exhibited some of these backlund transformations are not injective therefore additional non commutative symmetry groups are found for some equations these non commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra a 1 1 new completely integrable third order integro differential equations some depending explicitly on x are given these new equations give rise to nonin equation connections between the singularity equations from the painleve analysis and the nonlinear equations for interacting solitons are established a common approach to singularity analysis and soliton structure is introduced the painleve analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine gordon and nonlinear schrodinger equation a method to recover the painleve series from its constant level term is exhibit the soliton singularity transform is recognized to be connected to the mobius group this gives rise to a darboux like result for the spectral properties of the recursion operator these connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton singularity transform this then leads to the result that only for scaling degrees 1 and 2 the usual painleve analysis can be carried out a new invariance principle connected to kernels of differential operators is discovered this new invariance for example connects the explicit solutions of the liouville equation with the miura transform simple methods are exhibited which allow to compute out of n soliton solutions of the kdv bargman potentials explicit solutions of equations like the harry dym equation certain solutions are plotted

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

the painleve classification dominant truncations and resonance analysis a classification of ode from the point of view of singular point analysis is suggested the general formula for the painleve resonances is derived and the resonance analysis of low order dominant truncations is performed the resonance formulae for the painleve chains are presented and the significance of the painleve chains for the classification is explained

nonlinear evolution equations of hamiltonian form and the painleve analysis a number of necessary conditions for a class of nonlinear partial differential equations to pass the painleve test with the kruskal ansatz is given a theory for the admissible resonance patterns of equations in hamiltonian form is developed and discussed for some important special cases based on these results on algorithm can be described which constructs all nonlinear evolution equations of normal type and a certain hamiltonian form

painleve singularity structure analysis of three component gross pitaevskii type equations in this paper we have studied the integrability nature of a system of three coupled gross pitaevskii type nonlinear evolution equations arising in the context of spinor bose einstein condensates by applying the painleve singularity structure analysis we show that only for two sets of parametric choices corresponding to the known integrable cases the system passes the painleve test

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

the interaction processes of the n soliton solutions for an extended generalization of vakhnenko equation the n soliton solutions for an extended generalization of vakhnenko equation are calculated by applying the improved hirota method and the variable transformations the n soliton solutions can be expressed explicitly furthermore the interaction processes are discussed for the n soliton solutions

traveling wave solutions for nonlinear equations using symbolic computation abstract in this paper a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations pdes is presented and implemented in a computer algebraic system the key idea of this method is to take full advantage of a riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh method it is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions in this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods more importantly for some equations with no extra effect we also find other new and more general solutions at the same time by introducing appropriate transformations our method is further extended to the nonlinear pdes whose balancing numbers may be any nonzero real numbers the efficiency of the method can be demonstrated on a large variety of nonlinear pdes such as those considered in this paper burgers huxley equation coupled korteweg de vries equation caudrey dodd gibbon kawada equation active dissipative dispersive media equation generalized fisher equation and nonlinear heat conduction equation

exact solutions of the complex modified korteweg de vries equation firstly some similarity reductions of the complex modified korteweg de vries equation cmkdv which arises in the asymptotic interpretation of one dimensional plane wave propagation in a quadratic micropolar medium are discussed although it is not a soliton equation solvable by inverse scattering transformation its similarity reductions obtained by the use of lie group methods are of mathematical interest secondly the painleve analysis developed by weiss et al 1983 for nonlinear partial differential equations is applied to the cmkdv equation and the data obtained by the truncation technique yield some analytical solutions of the ordinary modified korteweg de vries equation and travelling wave solutions of the cmkdv equation which are also solutions of the similarity reduction obtained by classical lie group analysis

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

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

clustered gauss huard algorithm for the solution of ax b abstract in this paper we present and evaluate a new parallel algorithm for gauss huard method with scaled partial pivoting strategy for the solution of linear systems of equations the algorithm is generic in the sense that matrix distribution methods are decoupled from the algorithm details which makes it applicable to a wide spectrum of distribution functions and hence saving the additional efforts that would be needed to develop a new algorithm for every instance of matrix distribution functions the proposed algorithm is first analyzed then evaluated using a cluster of networked workstations a timing model is developed and verified to accurately estimate the execution time of the proposed algorithm the obtained results reveal two important observations first cluster computing is a viable and low cost alternative for solving computationally intensive problems such as achieving fast and stable solutions for systems of linear equations a problem often encountered in many real life applications second the obtained results reveal that gauss huard performs better than the well known gauss jordan algorithm in cluster environments

a bidirectional kaup kupershmidt equation and directionally dependent solitons exact multisoliton solutions are obtained for the bidirectional kaup kupershmidt equation that was reported in a recent paper j math phys 42 2567 2001 this fills the gap that was left in the prior study these solutions are found to possess the same remarkable and unusual property as the solitary wave solution namely their wave profiles are directionally dependent in particular solitons are presented which describe multiple head on collisions between single and double humped solitary waves explicit expressions are given for the first few multisoliton solutions and a procedure is described for constucting the general n soliton by iteration on the solitons of lower order

on soliton solutions of the kaup kupershmidt equation i direct bilinearisation and solitary wave abstract in this first of two papers designated i and ii physica d 137 2000 34 a direct procedure is given for bilinearising the kaup kupershmidt and sawada kotera equations it is found that whereas the sawada kotera equation can be written in bilinear form in two distinct ways which are shown to be equivalent the kaup kupershmidt equation has a single bilinear representation this distinguishing feature which has previously received no explanation has consequences for obtaining the soliton solutions of these two closely related fifth order integrable equations the anomalous solitary wave of the kaup kupershmidt equation is reconstructed and a simplified approach to obtaining its n soliton solutions is proposed this provides the basis for constructing these solutions explicitly in ii

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

the hirota s bilinear method and the tanh coth method for multiple soliton solutions of the sawada kotera kadomtsev petviashvili equation abstract in this work we derive a new completely integrable dispersive equation the equation is obtained by combining the sawada kotera sk equation with the sense of the kadomtsev petviashvili kp equation the newly derived sawada kotera kadomtsev petviashvili sk kp equation is studied by using the tanh coth method to obtain single soliton solution and by the hirota bilinear method to determine the n soliton solutions the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

polytopal linear groups the main objects of this paper are the graded automorphisms of w x polytopal semigroup rings i e semigroup rings k s where k is a field p and s is the semigroup associated with a lattice polytope p bruns p w x w x gubeladze and trung bgt the generators of k s correspond bijecp tively to the lattice points in p and their relations are the binomials representing the affine dependencies of the lattice points the simplest examples of such rings are the polynomial rings w x k x x with the standard grading they are associated to the unit 1 n w x simplices d the graded automorphism group gl k of k x x ny1 n 1 n is generated by diagonal and elementary automorphisms our main result is a generalization of this classical fact to the graded automorphism group g p of an arbitrary polytopal semigroup ring it says that each automork phism g g g p has a non unique normal form as a composition of toric k and elementary automorphisms and symmetries of the underlying poly tope in view of this analogy we call the groups g p polytopal linear k groups as in the case of gl k the elementary and toric automorphisms n generate g p if and only if this group is connected the elementary k automorphisms are defined in terms of so called column structures on p

backlund transformation lax pair and solutions for the caudrey dodd gibbon equation by using bell polynomials and symbolic computation we investigate the caudrey dodd gibbon equation analytically through a generalization of bells polynomials its bilinear form is derived based on which the periodic wave solution and soliton solutions are presented and the soliton solutions with graphic analysis are also given furthermore backlund transformation and lax pair are derived via the bells exponential polynomials finally the ablowitz kaup newell segur system is constructed

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

rehabilitation of the gauss jordan algorithm in this paper a gauss jordan algorithm with column interchanges is presented and analysed we show that in contrast with gaussian elimination the gauss jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability for solutions obtained by gauss jordan with column interchanges a more satisfactory bound for the residual norm can be given the analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by gaussian elimination and that in most practical situations the residuals are equally small this is confirmed by numerical experiments moreover timing experiments on a cyber 205 vector computer show that the algorithm presented has good vectorisation properties

painleve chains for the study of integrable higher order differential equations the concept of painleve chains is extended to chains of ordinary differential equations obtained by successive and simultaneous differentiation of both sides of equations of the general type f chi u ux u2x unx g chi u ux u2x umx where m or n 1 the three painleve chains are thus generated by such successive differentiation of u2x ku2 kdv chain u2x ku3 modified kdv chain and ux ku2 burgers chain hybrid painleve chains can be analogously obtained by successive differentiation of hybrid differential equations such chains obtained from u2x uux u3 and u2x 3uux u3 are described in detail passive differential equations in which the balancing exponent must depend upon the coefficients can also lead analogously to painleve chains as illustrated by those obtained from u2x ku2x u where k 2 or 3 2 the schwarzian derivative u3x ux 3 2 u2x ux 2 generates a painleve chain in which the members have consecutive integral resonances starting with the ubiquitous 1 chains of higher order differential equations exhibiting some but not all of the features of painleve chains can be obtained from the second order eigenvalue problem the dominant truncations of most evolution equations as well as the painleve canonical equations including the irreducible painleve transcendents appear in painleve chains generated these methods

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

parallel algorithms for solving large linear systems the solution of linear systems continues to play an important role in scientific computing the problems to be solved often are of very large size so that solving them requires large computer resources to solve these problems at least supercomputers with large shared memory or massive parallel computer systems with distributed memory are needed this paper gives a survey of research on parallel implementation of various direct methods to solve dense linear systems in particular are considered gaussian elimination gauss jordan elimination and a variant due to huard 19791 and an algorithm due to enright 19781 designed in relation to solving stiff odes such that stepsize and other method parameters can easily be varied some theoretical results are mentioned including a new result on error analysis of huard s algorithm moreover practical considerations and results of experiments on supercomputers and on a distributed memory computer system are presented

implementing linear algebra algorithms for dense matrices on a vector pipeline machine this paper examines common implementations of linear algebra algorithms such as matrix vector multiplication matrix matrix multiplication and the solution of linear equations the different versions are examined for efficiency on a computer architecture which uses vector processing and has pipelined instruction execution by using the advanced architectural features of such machines one can usually achieve maximum performance and tremendous improvements in terms of execution speed can be seen over conventional computers

cohomology of partially ordered sets and local cohomology of section rings abstract we study local cohomology of rings of global sections of sheafs on the alexandrov space of a partially ordered set we give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks the face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan thus we obtain a decomposition of the local cohomology of such face rings since the stanley reisner ring of a simplicial complex is the face ring of a rational pointed fan our main result can be interpreted as a generalization of hochster s decomposition of local cohomology of stanley reisner rings

complete toric varieties with reductive automorphism group we give equivalent and sufficient criteria for the automorphism group of a complete toric variety respectively a gorenstein toric fano variety to be reductive in particular we show that the automorphism group of a gorenstein toric fano variety is reductive if the barycenter of the associated reflexive polytope is zero furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of demazure roots

systematics of strongly self dominant higher order differential equations based on the painleve analysis of their singularities this paper presents a simple way of classifying higher order differential equations based on the requirements of the painleve property i e the presence of no movable critical points the fundamental building blocks for such equations may be generated by strongly self dominant differential equations of the type x nu xm u m n p p in which m and n are positive integers and p is a negative integer such differential equations having both a constant degree d and a constant value of the difference n m form a painleve chain however only three of the many possible painleve chains can have the painleve property among the three painleve chains that can have the painleve property one contains the burgers equation another contains the dominant terms of the first painleve transcendent the isospectral korteweg de vries equation and the isospectral boussinesq equation and the third contains the dominant terms of the second painleve transcendent and the isospectral modified cubic korteweg de vr