on the waterbag model of the dispersionless kp hierarchy ii we construct the bi hamiltonian structure of the waterbag model of dkp and establish the third order hamiltonian operator associated with the waterbag model also the symmetries and conserved densities of the rational type are discussed

a note on reductions of the dispersionless toda hierarchy we investigate the algebraic structure of the dispersionless hirota equations for the m 1 variable reductions of the dispersionless toda hierarchy which contains the special case for m 1 considered by kodama and pierce we demonstrate the m 2 case to illustrate the obtained result

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

on dispersionless lax representations for two primary models the equivalence between bi hamiltonian formulation and lax representation of dispersionless integrable hierarchies associated with two primary models of frobenius manifolds is explicitly verified

block type symmetry of bigraded toda hierarchy in this paper we define orlov schulman s operators ml mr and then use them to construct the additional symmetries of the bigraded toda hierarchy we further show that these additional symmetries form an interesting infinite dimensional lie algebra known as a block type lie algebra whose structure theory and representation theory have recently received much attention in the literature by acting on two different spaces under the weak w constraints we find in particular two representations of this block type lie algebra

solving the dispersionless dym equation by hodograph transformations the method of hodograph transformation is used to find a class of exact solutions of the dispersionless dym ddym equation and its reductions in particular we discuss hodograph solutions of one variable and two variable reductions of the ddym system and investigate their associated lax as well as hamiltonian formalism

darboux egoroff metrics rational landau ginzburg potentials and the painleve vi equation we present a class of three dimensional integrable structures associated with the darboux egoroff metric and classical euler equations of free rotations of a rigid body they are obtained as canonical structures of rational landau ginzburg potentials and provide solutions to the painleve vi equation

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

on the benney hierarchy free energy string equation and quantization the bi hamiltonian structure of the benney hierarchy is revisited we show that the compatibility condition of the poisson brackets provides the genus zero free energy of a topological field theory coupled to two dimensional gravity we calculate the correlation functions via the landau ginzburg formulation and derive the string equation based on the twistor construction moreover by using the approach of dubrovin and zhang we compute the genus one correction of the poisson brackets and compare them with the oevel strampp brackets of the kaup broer hierarchy

quasi classical dressing approach to the weakly dispersive kp hierarchy the recently proposed quasi classical dressing method provides a systematic approach to studying the weakly dispersive limit of integrable systems we apply the quasi classical dressing method to describe dispersive corrections of any order we show how to calculate the problems at any order for a rather general class of integrable systems presenting explicit results for the kp hierarchy case we demonstrate the stability of the method at each order we construct an infinite set of commuting flows at first order which allows a description analogous to the zero order purely dispersionless case highlighting a whitham type structure obstacles for the construction of the higher order dispersive corrections are also discussed

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

a new summation identity for the srivastava singhal polynomials abstract in his recent investigations involving differential operators for some generalizations of the classical laguerre polynomials h bavinck j phys a math gen 29 1996 l277 l279 encountered and proved a certain summation identity for the classical laguerre polynomials the main object of this sequel to bavinck s work is to prove a generalization of this summation identity for the srivastava singhal polynomials the demonstration which is presented here in the general case differs markedly from the earlier proof given for the known special case it is also indicated how the general summation identity can be applied to derive the corresponding result for one class of the konhauser biorthogonal polynomials

some polynomial systems associated with a certain family of differential operators the authors aim at presenting several presumably new classes of linear bilinear and mixed multilateral generating functions for some general systems of polynomials which are defined by means of a certain family of differential operators some of the generating functions considered here are associated with the stirling numbers of the second kind many known or new consequences and applications of the results obtained in this paper are also indicated

some families of hypergeometric polynomials and associated integral representations the main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single double and triple integral representations some known or new consequences of the general results presented here involving such classical orthogonal polynomials as the jacobi laguerre hermite and bessel polynomials and various other relatively less familiar hypergeometric polynomials are also considered each of the integral representations which are derived in this paper may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials

lowner equations hirota equations and reductions of the universal whitham hierarchy this paper reconsiders finite variable reductions of the universal whitham hierarchy of genus zero in the perspective of dispersionless hirota equations in the case of one variable reduction dispersionless hirota equations turn out to be a powerful tool for understanding the mechanism of reduction all relevant equations describing the reduction lowner type equations and diagonal hydrodynamic equations can be thereby derived and justified in a unified manner the case of multi variable reductions is not so straightforward nevertheless the reduction procedure can be formulated in a general form and justified with the aid of dispersionless hirota equations as an application previous results of guil manas and martinez alonso are reconfirmed in this formulation

degenerate frobenius manifolds and the bi hamiltonian structure of rational lax equations the bi hamiltonian structure of certain multicomponent integrable systems generalizations of the dispersionless toda hierarchy is studied for systems derived from a rational lax function one consequence of having a rational rather than a polynomial lax function is that the corresponding bi hamiltonian structures are degenerate i e the metric that defines the hamiltonian structure has a vanishing determinant frobenius manifolds provide a natural setting in which to study the bi hamiltonian structure of certain classes of hydrodynamic systems some ideas on how this structure may be extended to include degenerate bi hamiltonian structures such as those given in the first part of the paper is given

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

classical r matrix like approach to frobenius manifolds wdvv equations and flat metrics a general scheme for the construction of flat pencils of contravariant metrics and frobenius manifolds as well as related solutions to witten dijkgraaf verlinde verlinde associativity equations is formulated the advantage is taken from the rota baxter identity and some relation being counterpart of the modified yang baxter identity from the classical r matrix formalism the scheme for the construction of frobenius manifolds is illustrated on the algebras of formal laurent series and meromorphic functions on riemann sphere

hydrodynamic reductions of the dispersionless harry dym hierarchy we investigate the reductions of the dispersionless harry dym hierarchy to systems of finitely many partial differential equations these equations must satisfy the compatibility condition and they are diagonalizable and semi hamiltonian by imposing a further constraint the compatibility is reduced to a system of algebraic equations whose solutions are described

dispersionless scalar integrable hierarchies whitham hierarchy and the quasiclassical dressing method the quasiclassical limit of the scalar nonlocal problem is derived and a quasiclassical version of the dressing method is presented dispersionless kadomtsev petviashvili kp modified kp and dispersionless two dimensional toda lattice 2dtl hierarchies are discussed as illustrative examples it is shown that the universal whitham hierarchy is nothing but the ring of symmetries for the quasiclassical problem the reduction problem is discussed and in particular the d2dtl equation of b type is derived

on kernel formulas and dispersionless hirota equations of the extended dispersionless bkp hierarchy we derive dispersionless hirota equations of the extended dispersionless bkp edbkp hierarchy proposed by takasaki from the method of kernel formula provided by carroll and kodama moreover we verify associativity equations wdvv equations in the edbkp hierarchy from dispersionless hirota equations and give a realization of associative algebra with structure constants expressed in terms of the residue formula

finding mixed families of special polynomials associated with appell sequences abstract in this paper certain mixed special polynomial families associated with appell sequences are introduced and their properties are established further operational rules providing connections between these families and known special polynomials are established which are used to derive the identities and results for the members of these new families determinantal definitions of the polynomials associated with appell family are also derived the approach presented is general

recursion operators of some equations of hydrodynamic type we give a general method for constructing recursion operators for some equations of hydrodynamic type admitting a nonstandard lax representation we give several examples for n 2 and n 3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators we also discuss a reduction of n 1 systems to n systems of some new equations of hydrodynamic type

hamiltonian formalism of whitham type hierarchies and topological landau ginsburg models we show that the bi hamiltonian structure of the averaged gelfand dikii hierarchy is involved in the landau ginsburg topological models fora n series the casimirs for the first p b give the correct coupling parameters for the perturbed topological minimal model the correspondence coupling parameters primary fields is determined by the second p b the partition function at the tree level and the chiral algebra for lg models are calculated for any genusg

dressing operator approach to the toroidal model of higher dimensional dispersionless kp hierarchy based upon the dressing operator approach we investigate the twistor theoretical construction additional symmetries hodograph solutions and miura transformation for the toroidal model of higher dimensional dispersionless kp hierarchy introduced by takasaki

the approach to the dispersionless kp hierarchy the dispersionless limit of the scalar nonlocal problem is derived it is given by a special class of nonlinear first order equations a quasiclassical version of the dressing method is presented it is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of beltrami type equations the universal whitham hierarchy and in particular the dispersionless kp hierarchy turn out to be rings of symmetries for the quasiclassical problem pacs number 02 30 uk

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

some properties of generalized multiple hermite polynomials the purpose of this paper is to introduce and discuss a more general class of multiple hermite polynomials in this work the explicit forms operational formulas and a recurrence relation are obtained furthermore we derive several families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple hermite polynomials

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

operational methods and laguerre gould hopper polynomials abstract in this paper the authors introduce the laguerre gould hopper polynomials by combining the operational methods with the principle of monomiality generating function series definition differential equation and certain other properties of laguerre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae for these polynomials

generalized multiple integral representations for a large family of polynomials with applications this paper aims to provide a natural generalization and unification of a series of multiple integral representations for special classes of hypergeometric polynomials recently obtained by several authors this generalization is obtained by considering a very large family of hypergeometric polynomials the multiple integral representations given in this paper may be viewed as linearization relationship for the product of two different members of the associated family of hypergeometric polynomials

bilateral generating functions for the erkus srivastava polynomials and the generalized lauricella functions abstract in some recent investigations involving differential operators for the generalized lagrange polynomials chan et al w c c chan c j chyan h m srivastava the lagrange polynomials in several variables integral transforms spec funct 12 2001 139 148 encountered and proved a certain summation identity for the lagrange polynomials in several variables which are popularly known in the literature as the chan chyan srivastava polynomials in the present paper we investigate several families of bilateral generating functions for the erkus srivastava polynomials which are more general than the chan chyan srivastava polynomials and the srivastava daoust or generalized lauricella functions

summation formulae for gould hopper generalized hermite polynomials in this article we derive summation formulae for the gould hopper generalized hermite polynomials g n m x y further we derive the summation formulae for polynomials related to g n m x y as applications furthermore we establish a summation formula for m variable generalized hermite polynomials h n 1 2 m x 1 x 2 x m

product of two hypergeometric functions with power arguments abstract the hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt respectively the result is a power series in t whose coefficients are hypergeometric polynomials brychkov has extended the product formula such that in one argument the variable t can be squared we obtain a further extension such that in both arguments the variable t can be raised to positive integer powers

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

new generating functions for a class of generalized hermite polynomials the main object of this paper is to present several presumably new families of linear bilinear and mixed multilateral generating functions for a certain interesting generalization of the classical hermite and laguerre polynomials some of these generating functions are associated with the stirling numbers of the second kind numerous known or new consequences of the results derived here also considered

certain classes of generating functions for the jacobi and related hypergeometric polynomials abstract for a certain class of generalized hypergeometric polynomials the authors first derive a general theorem on bilinear bilateral and mixed multilateral generating functions and then apply these generating functions in order to deduce the corresponding results for the classical jacobi and laguerre polynomials they also consider several linear generating functions for these polynomials as well as for some multivariable jacobi and multivariable laguerre polynomials which were investigated in recent years some of the linear generating functions presented in this paper are associated with the stirling numbers of the second kind

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

a note on the lagrange polynomials in several variables abstract recently chan chyan and srivastava w c c chan c j chyan h m srivastava the lagrange polynomials in several variables integral transform spec funct 12 2001 139 148 introduced and systematically investigated the lagrange polynomials in several variables in the present paper we derive various families of multilinear and multilateral generating functions for the chan chyan srivastava multivariable polynomials

geometrical optics in nonlinear media and integrable equations it is shown that the geometrical optics limit of the maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of the dielectric constant on frequency and fields gives rise to the dispersionless veselov novikov equation for the refractive index it is demonstrated that the dispersionless veselov novikov hierarchy is amenable to the quasiclassical dressing method a connection is noted between the geometrical optics phenomena under consideration and the quasiconformal mappings on the plane

integral representations for the generalized bedient polynomials and the generalized cesaro polynomials abstract the main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations by suitably specializing our main results the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as for example the generalized bedient polynomials and the generalized cesaro polynomials each of the integral representations which are derived in this paper may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials

generalized polynomials operational identities and their applications abstract it is shown that an appropriate combination of methods relevant to generalized operational calculus and to special functions can be a very useful tool to treat a large body of problems both in physics and mathematics we discuss operational methods associated with multivariable hermite laguerre legendre and other polynomials to derive a wealth of identities useful in quantum mechanics electromagnetism optics etc or to derive new identities between special functions as e g mehler or mixed type generating functions

continuum analogues of contragredient lie algebras lie algebras with a cartan operator and nonlinear dynamical systems we present an axiomatic formulation of a new class of infinitedimensional lie algebras the generalizations ofz graded lie algebras with generally speaking an infinite dimensional cartan subalgebra and a contiguous set of roots we call such algebras continuum lie algebras the simple lie algebras of constant growth are encapsulated in our formulation we pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the cartan operator the generalization of the cartan matrix is a linear operator special examples of these algebras are the kac moody algebras algebras of poisson brackets algebras of vector fields on a manifold current algebras and algebras with differential or integro differential cartan operator the nonlinear dynamical systems associated with the continuum contragredient lie algebras are also considered

meromorphic lax representations of 1 1 dimensional multi hamiltonian dispersionless systems rational lax hierarchies introduced by krichever are generalized a systematic construction of infinite multi hamiltonian hierarchies and related conserved quantities is presented the method is based on the classical r matrix approach applied to poisson algebras a proof that poisson operators constructed near different points of laurent expansion of lax functions are equal is given all results are illustrated by several examples

integrable quasiclassical deformations of cubic curves a general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations

some generalizations of the euler knopp transformation the purpose of this paper is to construct a generalization of the euler knopp transformation using this one may recover previously known transformations derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well known functions

operational identities and properties of ordinary and generalized special functions the theory of hermite laguerre and of the associated generating functions is reformulated within the framework of an operational formalism this point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities in this paper a central role is played by negative derivative operators and by their link with the tricomi functions and the generalized laguerre polynomials

on the whitham hierarchy dressing scheme string equations and additional symmetries a new description of the universal whitham hierarchy in terms of a factorization problem in the lie group of canonical transformations is provided this scheme allows us to give a natural description of dressing transformations string equations and additional symmetries for the whitham hierarchy we show how to dress any given solution and prove that any solution of the hierarchy may be undressed and therefore comes from a factorization of a canonical transformation a particularly important function related to the tau function appears as a potential of the hierarchy we introduce a class of string equations which extends and contains previous classes of string equations considered by krichever and by takasaki and takebe the scheme is also applied for a convenient derivation of additional symmetries moreover new functional symmetries of the zakharov extension of the bentley gas equations are given and the action of additional symmetries over the potential in terms of linear pdes is characterized

some properties of legendre gould hopper polynomials and operational methods abstract in this paper the author introduces the legendre gould hopper polynomials by combining the operational methods with the principle of monomiality generating functions series definition differential equation and certain other properties of legendre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae certain summation formulae for these polynomials are also obtained

some families of generating functions associated with the stirling numbers of the second kind the object of this paper is to present a systematic introduction to and several interesting applications of a general result on generating functions associated with the stirling numbers of the second kind for a fairly wide variety of special functions and polynomials in one two and more variables the main results given below are shown to apply not only to the classical orthogonal polynomials including for example the jacobi polynomials which contain as their special cases the gegenbauer or ultraspherical polynomials the legendre or spherical polynomials and the chebyshev polynomials of the first and second kinds and the laguerre polynomials and to their various extensions and generalizations studied in recent years but indeed also to a class of generalized hypergeometric functions the lauricella polynomials in several variables and the familiar lagrange polynomials which arise in certain problems in statistics relevant connections of some of these families of generating functions with various known results are also indicated

on a problem of common approximate fixed points the nonstandard analysis methods are used to propose some new ideas concerning the problem of common approximate fixed points for a commuting family of nonexpansive mappings

tau function and hirota bilinear equations for the extended bigraded toda hierarchy in this paper we generalize the sato theory to the extended bigraded toda hierarchy we revise the definition of the lax equations give the sato equations wave operators hirota bilinear identities and show the existence of tau function t meanwhile we prove the validity of its fay like identities and hirota bilinear equations hbes in terms of vertex operators whose coefficients take values in the algebra of differential operators in contrast to hbes of the usual integrable system the current hbes are equations of the product of operators involving e x and t

singular sector of the burgers hopf hierarchy and deformations of hyperelliptic curves we discuss the structure of shock singularities of the burgers hopf hierarchy it is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy the stratification is associated with the birkhoff decomposition of the grassmannian given by the set of linear spaces spanned by the hierarchy we then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum the hierarchy is called the hidden burgers hopf hierarchy and we found the riemann invariant form and the hodograph solution

integrability properties of the dispersionless kadomtsev petviashvili hierarchy in the paper we investigate integrability characteristics for the dispersionless kadomtsev petviashvili hierarchy these characteristics include symmetries hamiltonian structures and conserved quantities we give a lax triad to construct a master symmetry and a hierarchy of non isospectral dispersionless kadomtsev petviashvili flows these non isospectral flows together with the known isospectral dispersionless kadomtsev petviashvili flows form a lie algebra which is used to derive two sets of symmetries for the isospectral dispersionless kadomtsev petviashvili hierarchy by means of the master symmetry symmetries noether operator and conserved covariants hamiltonian structures are constructed for both isospectral and non isospectral dispersionless kadomtsev petviashvili hierarchies finally two sets of conserved quantities and their lie algebra are derived for the isospectral dispersionless kadomtsev petviashvili hierarchy

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

neocompact sets and the fixed point property abstract we use a newly introduced concept of neocompactness to study problems from metric fixed point theory in particular we give a sufficient condition for a superreflexive banach space x to have the fixed point property and obtain shorter proofs of some well known results in that theory