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

a higher order triangular plate bending element revisited a new formulation of an eighteen degrees of freedom higher order triangular plate bending element using triangular area co ordinates is presented the displacement function w is taken as the complete fifth order polynomial in area co ordinates the normal slope along an edge of the triangle is constrained to vary cubically the twenty one constants are expressed explicity in terms of eighteen degrees of freedom the element stiffness matrix is expressed as a product of component matrices for which explicit expressions are developed and presented no numerical inversion or integration is necessary the formulation is expected to be useful specially for microcomputers

non isospectral scattering problems and truncation for hierarchies burgers and dispersive water waves abstract non isospectral scattering problems have been proven useful for several reasons amongst them the information that they provide about painleve truncation for entire hierarchies of integrable partial differential equations pdes we show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 1 dimensions namely burgers hierarchy and the dispersive water wave hierarchy burgers equation is well known as a model for turbulence and the dispersive water wave equations as a system governing shallow water waves we also see how these results are easily extended to related hierarchies in 2 1 dimensions our results for the 2 1 dimensional burgers hierarchy and for the 1 1 and 2 1 dimensional dispersive water wave hierarchies are all new

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

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

modified extended tanh function method and its applications to nonlinear equations new exact travelling wave solutions for the generalized shallow water wave equation the improved boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh function method the obtained results include rational periodic singular and solitary wave solutions

stress analysis of flat plates with shear using explicit stiffness matrix an explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation the element has twelve nodes on the sides and four nodes internal to it the formulation is displacement type and the use of area co ordinates makes it possible to obtain the shape functions explicitly separate polynomials are assumed for transverse displacement and rotations to obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary the element which is initially of thirty five degrees of freedom can be reduced to a thirty degrees of freedom one by condensation of the internal nodes an interesting feature of the element developed is that the values of nodal moments computed at a node point considering different elements surrounding the node do not vary significantly thus the nodal moments can be obtained directly at node points also the element does not give rise to any inconvenience like locking even for very thin plates the straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less cpu time thus making the use of the element well suited to low capacity computers a number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results good accuracy has been observed in all cases even for a small number of elements

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

a new general algebraic method with symbolic computation to construct new doubly periodic solutions of the 2 1 dimensional dispersive long wave equation for constructing more new exact doubly periodic solutions in terms of rational form jacobi elliptic function of nonlinear evolution equations a new direct and unified algebraic method named jacobi elliptic function rational expansion method is presented and implemented in a computer algebraic system compared with most of the existing jacobi elliptic function expansion methods the proposed method can be expected to obtain new and more general formal solutions we choose a 2 1 dimensional dispersive long wave equation to illustrate the method

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

generating difference equations with the darboux transformation starting with a system satisfying the so called bispectral property a method is given to generate other systems with still higher order recursion relations the usefulness of the method is checked for the classical examples

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

a classification of 2 simple prehomogeneous vector spaces of type i let p g gl v be a rational representation of a connected linear algebraic group g on a finite dimensional vector space v all defined over an algebraically closed field k of characteristic zero if v has a zariskidense g orbit we call a triplet g p v a prehomogeneous oector space abbrev p v when p is irreducible such p v s have been classified in 11 since then it has turned out gradually that the complete classification of reductive p v s i e p v s with reductive groups g is an extremely laborious task therefore it is natural to classify some restricted class of p v s e g 2 to get some insight into the general situation a p v g p v is called a 2 simple p v when 1 g gl 1 x g x g with simple algebraic groups g and gz 2 p is the composition of a rational representation p of g x g of the form p p1 0 p pkol a a l l z t with k s t l where pi tv resp pi t are nontrivial irreducible representations of g resp g and the scalar multiplications gz 1 on each irreducible component v for i 1 f where v i v we say that a 2 simple p v g p v is of type zif k l and at least one of gl l xg xg pi pi i 1 k is a nontrivial p v see definition 5 p 43 in 11 on the other hand if k l and all gl l xg xg p op i l k are trivial p v s it is called a 2 simple p v of type ii in 3 all 2 simple p v s of type ii has been already classified in this paper we shall classify all 2simple p v s of type i thus together with 3 we complete a classification of all 2 simple p v s for example the fact that all irreducible p v s are castling equivalent to 2 simple p v s or to x m x y m x g 2 1 1 i i n 1 with m 2 3 see 1 indicates the importance of 2 simple p v s for simplicity we write g p v or g p instead of g p v

on soliton type solutions of equations associated with n component systems the algebraic geometric approach to n component systems of nonlinear integrable pde s is used to obtain and analyze explicit solutions of the coupled kdv and dym equations detailed analysis of soliton fission kink to anti kink transitions and multi peaked soliton solutions is carried out transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics

numerical analysis of thick plates bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory the theory is based on a higher order displacement model and the three dimensional hooke s laws for plate material giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness it is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency economy reliability and accuracy in such applications

arithmetic calculus of fourier transforms by igusa local zeta functions we show the possibility of explicit calculation of the fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over jr by using the explicit form of p adic igusa local zeta functions let g p v be a regular prehomogeneous vector space defined over r and fi f the basic r relative invariants of g p v cf f sato 2 p 444 for explicit calculation of fourier transforms of the complex power r if x is 1 if1 x isi i 1 with s si sr e cr two general methods are known when g p v has finitely many g orbits one can use a microlocal calculus established by m kashiwara according to m sato s idea see p 404 in m k k m when r 1 and the relative invariant is linear for each variable one can use igusa s method see p 8 in igusa 2 in this paper we shall show the possibility of calculation for some cases by using the explicit form of the igusa s local zeta function based on the idea of iwasawa tate theory iwasawa tate as an example we shall calculate the fourier transform of if x is for gl4 x sl2m l a2 d al d a d a alt2m 1 d aff2m l d aff2m i 1 d aff2m which has infinitely many orbits and r 4 so that it has not been calculated by other methods the igusa local zeta function of this prehomogeneous vector space has been explicitly calculated by hosokawa our method is applicable for a prehomogeneous vector space with za rzm for some constant r 0 for example irreducible resp simple 2 simple of type i regular prehomogeneous vector spaces with finitely many adelic open orbits satisfy za rzm see igusa 4 and k k this paper has four sections 1 preliminaries 2 functional equation 3 unramified f factors and igusa local zeta functions 4 arithmetic calculus example the last section of this paper was completed when the author was received by the editors july 27 1993 and in revised form march 30 1994 1991 mathematics subject classification primary 32m1 2 20g20 secondary 1 1s40

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

a new property of reproducing kernels for classical orthogonal polynomials abstract we exhibit a second order differential operator commuting with the reproducing kernel n 0 t n n h n each time that n is one of the classical orthogonal polynomials jacobi laguerre hermite and bessel this is the analog of a known property in the study of time and band limited signals

free vibration analysis of isotropic rectangular plates using a high order triangular finite element with shear in this paper free vibration analysis of isotropic rectangular plates with different thickness ratios different boundary conditions and different aspect ratios has been investigated using a high order triangular element the first order shear deformation theory fosdt is used to include the effect of transverse shear deformation the element has 18 nodes on the sides and six internal nodes the geometry of the element is expressed by three linear shape functions of area coordinates the formulation is displacement type and the use of area coordinates makes the shape functions for field variables to be expressed explicitly no numerical integration is required to get the element stiffness and consistent mass matrices the element has 51 degrees of freedom which can be reduced to 39 degrees of freedom by guyan reduction mass condensation or eigenvalue economization scheme for the degrees of freedom associated with the internal nodes rotary inertia has been included in the consistent mass matrix numerical examples are presented to show the accuracy and convergence characteristics of the element

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions abstract starting from the hyperoctahedral multivariate hypergeometric function of heckman and opdam associated with the b c n root system we arrive via partial confluent limits in the sense of oshima and shimeno at solutions of the eigenvalue equations for the quantum toda chain with one sided boundary perturbations of poschl teller type and for the hyperbolic quantum calogero sutherland system in a morse potential with the aid of corresponding degenerations of the bispectral dual difference equations for the heckman opdam hyperoctahedral hypergeometric function it is deduced that the eigensolutions in question are holomorphic in the spectral variable

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

finite band solutions of the classical boussinesq burgers equations the classical boussinesq burgers hierarchy is decomposed into two systems of solvable ordinary differential equations with the help of the lax representations of stationary evolution equations from which the finite band solutions of the higher order classical boussinesq burgers equations are obtained

commuting linear differential expressions we consider the question when do two ordinary linear differential expressions commute it turns out that the set of all expressions which commute with a given one form a commutative ring here we study the algebraic structure of these rings as an application a complete characterization of normal differential expressions is obtained

on a direct bilinearization method kaup s higher order water wave equation as a modified nonlocal boussinesq equation a systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of bell s exponential polynomials 1934 application of this procedure to kaup s higher order wave equation 1975 discloses several links with other soliton systems in particular it is found that the kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good boussinesq equation

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

modified singular manifold expansion application to the boussinesq and mikhailov shabat systems we present a modified treatment of the singular manifold method as an improved variant of the painleve analysis for partial differential equations with two branches in the painleve expansion we illustrate the method by fully applying it to the classical boussinesq system and to the mikhailov shabat system

conslaw a maple package to construct the conservation laws for nonlinear evolution equations an improved algorithm to explicitly compute the polynomial type conservation laws for nonlinear evolution equations either uniform in rank or not is introduced and a software package conslaw written in maple to automate the computation is developed conslaw can construct the polynomial type conservation laws for polynomial partial differential equations automatically furthermore some new integrable systems can be filtered out by analyzing the compatibility conditions which guarantee the existence of the conservation laws for a given parameterized nonlinear evolution equations the explicit forms of the conserved densities play an important role in studying the integrability such as explicit analytical solutions bi hamiltonian form one parameter family of backlund transformations lax pairs and the checking of the accuracy of numerical integration algorithm the effectiveness of conslaw is illustrated by applying it to a variety of equations

transformation formulae of the solutions of the two dimensional toda lattice two transformation formulae of the solutions of the two dimensional toda lattice are presented the first formula which can be obtained by the zakharov shabat dressing method is proved by applying the hirota bilinear transformation method 1988 and gives the basis for the second new formula

some p v equivalences and a classification of 2 simple prehomogeneous vector spaces of type a classification of 2 simple prehomogeneous vector spaces is completed by using some p v equivalences together with 3 some part is very different from the previous classification of the irreducible or simple cases 1 2 and some new method is necessary this result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces introduction in m sato and t kimura 1 all irreducible prehomogeneous vector spaces abbreviated irreducible p v s are classified up to castling equivalence and it is proved that any irreducible p v is castling equivalent to a 2 simple p v or to sl m x sl m x gl 2 a1 x a1 x a1 with m 2 3 therefore as a step to a classification of the general case it is natural to classify all nonirreducible 2 simple p v s in this paper we shall classify all 2 simple p v s of type ii and give the complete list of them up to strong equivalence in the sense of definition 4 p 36 in 1 however we need various p v equivalences cf propositions 1 7 1 12 1 34 1 36 1 37 theorem 1 16 and its corollary corollary 1 26 definition 1 31 remark 1 32 to carry out the classification and hence first we shall prove them we also investigate the regularity in some cases since all 2 simple p v s of type i are already classified in 3 this completes the classification of all 2 simple p v s here we say that g p v or simply g p is a 2 simple p v if it is a prehomogeneous vector space of the following form g gl 1 k s t xg x g2 where g1 and g2 are simple algebraic groups p the composition of the scalar multiplications gl l k s on each irreducible component and the representation p p1 p 1 pk p k a1 s 1 1 t1 t ofthegroupg1 x g2 where pi aj resp p tj are nontrivial irreducible representations of g1 resp g2 we write g p v instead of g p v for simplicity to avoid confusion we write instead of clearly each gl 1 x g1 x g2 pi p i 1 k is an irreducible 2 simple p v if at least one of them is a nontrivial p v see definition 1 1 we call it a 2 simple p v of type i on the other hand if all of them are trivial p v s we call it a 2 simple p v of type ii in this case we have g2 sl n received by the editors march 3 1986 and in revised form july 30 1986 1980 mathematics subject classification 1985 revision primary 11r20 11r29 secondary 11n25 11r18 01988 american mathematical society 0002 9947 88 1 00 25 per page

local well posedness for the super korteweg de vries equation abstract we establish some qualitative properties for the cauchy problem associated with the super korteweg de vries equation super kdv super kdv t u x 3 u 1 2 x u 2 1 2 x 2 v 2 0 t v x 3 v x u v 0 x t r we prove local well posedness in weighted sobolev spaces x s 3 h s r h 3 x 2 d x h s r h 3 x 2 d x s 5 integer provided that the initial data is small enough and also in x s 11 h s r h 11 x 2 d x h s r h 11 x 2 d x with s 15 integer for arbitrary initial data the main ingredients of the proof for the first case are new estimates describing the smoothing effect of kato type for the kdv group w t t r that for the second case is via a change of variable performed and a deduction of new smoothing effects related to the kdv c kenig g staffilani local well posedness for higher order nonlinear dispersive systems j fourier anal appl 3 4 1997

huygens principle for hyperbolic operators and integrable hierarchies abstract we show that the stationary solutions of the canonical akns hierarchy of nonlinear evolution equations yield perturbations of dirac operators that satisfy a strict form of huygens principle namely the domain of dependence of such dirac operators at any given point y is contained in the light cone s hypersurface issued from y by the canonical akns hierarchy we mean that the differential polynomials defining the flows are isobaric with respect to certain weights the method we employ is of interest by itself indeed we consider the riesz kernels associated to a given hyperbolic differential operator and expand the fundamental solution of perturbations of this operator in a series in such riesz kernels using the coefficients of this hadamard type expansion we introduce a family of vector fields for the d alembertian such vector field family corresponds to the kdv hierarchy and for the dirac operators they include the akns one

bilinearization of the non local boussinesq equation a single field bilinear system generating the so called non local boussinesq equation is constructed from the bilinearization procedure it can be seen that the associated hierarchy of soliton systems which we construct shares part of the solution set of a hierarchy related to the kadomtsev petviashvili equation a bilinear representation of the recursion operator for the kaup hierarchy is essential in the construction and a systematic way of obtaining such a representation from just two soliton considerations is presented

multiphase similarity solutions of integrable evolution equations abstract we introduce a new class of solutions to integrable nonlinear evolution equations which are thought to have a deep connection with the n point correlation functions of exactly solvable models in statistical mechanics

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

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 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