well quasi orders generated by a word shuffle rewriting given a set i of words the set l i e of all words obtained by the shuffle of copies of words of i is naturally provided with a partial order for u v in l i e u i v if and only if v is the shuffle of u and another word of l i e in f d alessandro s varricchio well quasi orders unavoidable sets and derivation systems in word avoidability complexity and morphisms wacam rairo theoretical informatics and applications 40 3 2006 407 426 special issue the authors have opened the problem of the characterization of the finite sets i such that i is a well quasi order on l i e in this paper we give an answer in the case when i consists of a single word w

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

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

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

a note on the generalized symmetries of lattice hierarchies we show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential delay equations all members of these hierarchies of lattice and differential delay equations commute it is thus seen that differential delay hierarchies provide a broader context within which to place lattice hierarchies

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

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

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

a nonisospectral extension of the volterra hierarchy to 2 1 dimensions we give a new nonisospectral generalization of the volterra lattice equation to 2 1 dimensions we use this to construct a new nonisospectral lattice hierarchy in 2 1 dimensions along with its underlying linear problem reductions yield a variety of new integrable hierarchies including generalizations of known discrete painleve hierarchies all along with their corresponding linear problems this represents an extension of previously developed techniques to the discrete case

nonisospectral scattering problems and similarity reductions abstract we give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems special cases of these changes of variables involve similarity variables and similarity solutions of a hierarchy are seen to correspond to time independent solutions of an equivalent hierarchy we thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same painleve hierarchies as examples we consider the korteweg de vries hierarchy and the dispersive water wave hierarchy

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

the kontsevich matrix integral convergence to the painleve hierarchy and stokes phenomenon we show that the kontsevich integral on n times n matrices n infty is the isomonodromic tau function associated to a 2 times 2 riemann hilbert problem the approach allows us to gain control of the analysis of the convergence as n to infty by an appropriate choice of the external source matrix in kontsevich s integral we show that the limit produces the isomonodromic tau function of a special tronquee solution of the first painleve hierarchy and we identify the solution in terms of the stokes data of the associated linear problem we also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal witten kontsevich tau function is the asymptotic expansion of each of them in their respective sectors thus providing an analytic tool to analyze its nonlinear stokes phenomenon

new integrable equations of fourth order and higher degree related to cosgrove s equation we give a general formulation of the algorithm of fokas and ablowitz which then allows us to obtain transformations for nth order ordinary differential equations to equations of the same order but perhaps of higher degree previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by painleve and co workers either to other equations in the painleve classification or to equations of second order and second degree as an example of our approach we consider a new fourth order ordinary differential equation due to cosgrove which is believed to define a new transcendent we obtain transformations relating this equation to other fourth order ordinary differential equations of degrees 2 all of these transformations as well as the corresponding higher degree differential equations all of which have the painleve property are new

symmetries of a 2 1 dimensional breaking soliton equation infinitely many symmetries for a 2 1 dimensional breaking soliton equation are constructed via the infinitesimal version of the dressing method these symmetries are proved to constitute an infinite dimensional lie algebra which contains some abelian and virasoro subalgebras the hierarchies of equations generated by these symmetries are also considered these hierarchies of equations are proved to be associated with the isospectral and non isospectral deformations of the akns spectral problem

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

antiquantization isomonodromy and integrability an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear

universality for the focusing nonlinear schrodinger equation at the gradient catastrophe point rational breathers and poles of the tritronquee solution to painleve i the semiclassical zero dispersion limit of solutions to the one dimensional focusing nonlinear schrodinger equation nls is studied in a scaling neighborhood d of a point of gradient catastrophe x 0 t 0 we consider a class of solutions specified in the text that decay as the neighborhood d contains the region of modulated plane wave with rapid phase oscillations as well as the region of fast amplitude oscillations spikes in this paper we establish the following universal behaviors of the nls solutions q near the point of gradient catastrophe i each spike has height and uniform shape of the rational breather solution to the nls scaled to the size ii the location of the spikes is determined by the poles of the tritronquee solution of the painleve i p1 equation through an explicit map between d and a region of the painleve independent variable iii if but lies away from the spikes the asymptotics of the nls solution is given by the plane wave approximation with the correction term being expressed in terms of the tritronquee solution of p1 the relation with the conjecture of dubrovin grava and klein about the behavior of solutions to the focusing nls near a point of gradient catastrophe is discussed we conjecture that the p1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point our technique is based on the nonlinear steepest descent method for matrix riemann hilbert problems and discrete schlesinger isomonodromic transformations 2013 wiley periodicals inc

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

on well quasi orders of free monoids abstract the paper investigates down sets associated to well quasi orders of particular language theoretic interest is the quasi order u s v resp u v of u being a subword resp a parikh subword of v as well as their inverses we establish a number of results about the regularity and effective regularity of the down sets in particular a general condition for the effective regularity of the down set associated to an inverse well quasi order theorem 7 1 it is decidable whether or not an arbitrary regular language results as a down set from an infinite or bi infinite word whereas the same problem is undecidable for context free language a quasi order being a well quasi order is connected to arbitrary languages being confluent

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

a note on the lax pairs for painleveequations for the classical painleveequations besides the method of similarity reduction of lax pairs for integrable partial differential equations two ways are known for lax pair generation the first is based on the confluence procedure in fuchs linear ode with four regular singularities isomonodromy deformation which is governed by the sixth painleveequation the second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of odes in whose non triangular extensions give rise to the lax pairs for the painleveequations the theory of integrable integral operators suggests a new way of lax pair generation for the classical painleveequations this method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions some of the lax pairs we introduce are known others are new the question of gauge equivalence of different lax pairs for the painleveequations is considered as well

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

on an extended second painleve hierarchy abstract we present a new extension of the second painleve hierarchy and study its properties in addition to lax pairs backlund transformations auto backlund transformations and basic special integrals we also consider a new phenomenon whereby we obtain relations between systems of different orders but of the same form the extension made here of the second painleve hierarchy is based on the use of non isospectral scattering problems and so is quite general we thus expect to be able to obtain similar extensions of other painleve hierarchies including not only for continuous examples but also for discrete and differential delay examples we believe that our work is also of relevance for painleve classification since it gives information about classes of equation that may be of interest and in addition provides a key to the possible identification of equations isolated in such a process

on the backlund transformation for the gel fand dickey equations we study the backlund transformation of the gel fand dickey equations and in particular how the factorization ofnth order differential operators leads to lax type equations for first order operators generalizing work of adler and moser 1 in a similar fashion we study the toda equations

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

differential calculi on some quantum prehomogeneous vector spaces this paper is devoted to study of differential calculi over quadratic algebras which arise in the theory of quantum bounded symmetric domains we prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k forms are the same as in the classical case this result is well known for quantum matrices the quadratic algebras which we consider in the present paper are q analogs of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type this enables us to prove that the de rham complex is isomorphic to the dual of a quantum analog of the generalized bernstein gelfand gelfand resolution lie goups and their representations summer school of the bolyai janos mathematical society edited by i m gelfand halsted new york 1975 pp 21 64

on the meromorphic solutions of an equation of hayman abstract the behavior of meromorphic solutions of differential equations has been the subject of much study research has concentrated on the value distribution of meromorphic solutions and their rates of growth the purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi parameter ordinary differential equation introduced by hayman this equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions only those that are meromorphic this is achieved by combining wiman valiron theory and local series analysis hayman conjectured that all entire solutions of this equation are of finite order all meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one

higher order painleve equations and their symmetries via reductions of a class of integrable models higher order painleve equations and their symmetry transformations belonging to extended affine weyl groups a 1 n are obtained through a self similarity limit of a class of pseudo differential lax hierarchies with symmetry inherited from the underlying generalized volterra lattice structure in particular an explicit example of the painleve v equation and its backlund symmetry is obtained through a self similarity limit of a generalized kdv hierarchy from aratyn et al 1995 int j mod phys a 10 2537

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

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

symmetries exact solutions and nonlinear superposition formulas for two integrable partial differential equations we recently introduced two new sixth order partial differential equations pdes associated with third order scattering problems here we extend our study of these pdes by considering the construction of exact solutions both by using the method of symmetry reduction due to lie and by using their darboux transformations dts amongst the ordinary differential equations odes obtained by symmetry reduction is an ode due to cosgrove that is believed to define a new painleve transcendent this ode provides soliton solutions for our integrable pdes that include arbitrary functions of time the dts for our pdes allow the recovery of these solutions and in addition provide other solutions which are not associated with lie symmetries either classical or nonclassical we also consider the iteration of the corresponding backlund transformations bts for these pdes the theorem of permutability allows us to reduce this process of iterating the dt from one of solving a third order linear equation the spatial p

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

an isomonodromy cluster of two regular singularities we consider a linear 2 2 matrix ode with two coalescing regular singularities this coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ode in particular a zero distance limit for the ode exists the monodromy group of the limiting ode is calculated in terms of the original one this coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations in our main example the latter limit reads as p 6 p 5 where p n is the nth painleve equation we also discuss some general problems which arise while studying the above mentioned limits for the painleve equations

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

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

covariant q differential operators and unitary highest weight representations for u q su n n we investigate a one parameter family of quantum harish chandra modules of uqsl2n this family is an analog of the holomorphic discrete series of representations of the group su n n for the quantum group uqsun n we introduce a q analog of the wave operator a determinant type differential operator and prove certain covariance property of its powers this result is applied to the study of some quotients of the above mentioned quantum harish chandra modules we also prove an analog of a known result by j faraut and a koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

the jacobi polynomial ensemble and the painleve vi equation we consider the jacobi polynomial ensemble of n n random matrices we show that the probability of finding no eigenvalues in the interval 1 z for a random matrix chosen from the ensemble viewed as a function of z satisfies a second order differential equation after a simple change of variable this equation can be reduced to the okamoto jimbo miwa form of the painleve vi equation the result is achieved by a comparison of the tracy widom and the virasoro approaches to the problem which both lead to different third order differential equations the virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments which drastically simplifies the computations

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

stokes matrices and monodromy of the quantum cohomology of projective spaces in this paper we compute stokes matrices and monodromy of the quantum cohomology of projective spaces this problem can be formulated in a classical framework as the problem of computation of stokes matrices and monodromy of differential equations with regular and irregular singularities we prove that the stokes matrix of the quantum cohomology coincides with the gram matrix in the theory of derived categories of coherent sheaves we also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

on quasi orders of words and the confluence property we investigate the confluence property that is the property of a language to contain for any two words of it one which is bigger with respect to a given quasi order on the respective free monoid than each of the former two this property is investigated mainly for regular and context free languages as a consequence of our study we give an answer to an old open problem raised by haines concerning the effective regularity of the sets of subwords namely we prove that there are families with a decidable emptiness problem for which the regularity of the sets of subwords is not effective

the indefinite anti self dual metrics and the painleve equations we classify the su 2 invariant anti self dual metrics with a signature the metrics are specified by a solution of painleve vi v iii or ii moreover we show the geometric meaning of the metrics specified by each type of painleve function

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

lattice modified kdv hierarchy from a lax pair expansion we produce a hierarchy of integrable equations by systematically adding terms to the lax pair for the lattice modified kdv equation the equations in the hierarchy are related to one another by recursion relations these recursion relations are solved explicitly so that every equation in the hierarchy along with its lax pair is known

quantum painleve calogero correspondence the painleve calogero correspondence is extended to auxiliary linear problems associated with painleve equations the linear problems are represented in a new form which has a suggestive interpretation as a quantized version of the painleve calogero correspondence namely the linear problem responsible for the time evolution is brought into the form of non stationary schrodinger equation in imaginary time t 12 x2 v x t whose hamiltonian is a natural quantization of the classical calogero like hamiltonian h 12p2 v x t for the corresponding painleve equation in present paper we present explicit constructions for the first five equations from the painleve list

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight

on the central quadric ansatz integrable models and painleve reductions it was observed by tod 1995 class quantum grav 12 1535 47 and later by dunajski and tod 2002 phys lett a 303 253 64 that the boyer finley bf and the dispersionless kadomtsev petviashvili dkp equations possess solutions whose level surfaces are central quadrics in the space of independent variables the so called central quadric ansatz it was demonstrated that generic solutions of this type are described by painlev equations piii and pii respectively the aim of our paper is threefold 1 based on the method of hydrodynamic reductions we classify integrable models possessing the central quadric ansatz this leads to the five canonical forms including bf and dkp 2 applying the central quadric ansatz to each of the five canonical forms we obtain all painlev equations pi pvi with pvi corresponding to the generic case of our classification 3 we argue that solutions coming from the central quadric ansatz constitute a subclass of two phase solutions provided by the method of hydrodynamic reductions

hidden symmetry of the differential calculus on the quantum matrix space a standard bicovariant differential calculus on the quantum matrix space is considered our main result is proving that the module differential algebra is in fact a module differential algebra

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy