painleve property and constants of the motion of the complex lorenz model the results of the painleve analysis and of a search for constants of the motion of the complex lorenz model are presented

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

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

on the equation y2 x x2 p generators are found for the group of rational points on the title curve for all primes p 5 mod 8 less than 1 000 the rank is always i in accordance with conjectures of selmer and mordell some of the generators are rather large

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

on the quantization of a self dual integrable system in this note we apply canonical quantization to the self dual particle system describing the motion of poles to a higher rank solution of the kp hierarchy explicitly determining both the quantum hamiltonian and the wavefunction it is verified that the quantum hamiltonian is trivially bispectral that is that the wavefunction can be taken to be symmetric as predicted by a widely held hypothesis of mathematical physics

formal solutions to the kp hierarchy we find all formal solutions to the dependent kp hierarchy they are characterized by certain cauchy like data the solutions are found in the form of formal series for the tau function of the hierarchy and for its logarithm the f function an explicit combinatorial description of the coefficients of the series is provided

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

solutions in closed form and as power series to the real lorenz equations using the method of the lie theory of extended groups and for the parameter values b 1 and r 0 we construct explicitly the general exact solution to the real lorenz equations in terms of jacobian elliptic functions in the context of our approach further possible completely integrable cases of the lorenz system are discussed by considering the result of the painleve analysis for 1 b 2 and r 1 9 and negative values of r the latter case r 0 and 0 not following from the painleve test for other positive parameter values and in the form of appropriate power series we find some particular exact solutions which do not possess the painleve property

a quantum group structure in integrable conformal field theories we discuss a quantization prescription of the conformal algebras of a class ofd 2 conformal field theories which are integrable we first give a geometrical construction of certain extensions of the classical virasoro algebra known as classicalw algebras in which these algebras arise as the lie algebra of the second hamiltonian structure of a generalized korteweg de vries hierarchy this fact implies that thew algebras obtained as a reduction with respect to the nilpotent subalgebras of the kac moody algebra describe the integrability of a toda field theory subsequently we determine the coadjoint operators of thew algebras and relate these to classical yang baxter matrices the quantization of these algebras can be carried out using the concept of a so called quantum group we derive the condition under which the representations of these quantum groups admit a hilbert space completion by exploring the relation with the braid group then we consider a modification of the miura transformation which we use to define a quantumw algebra this leads to an alternative interpretation of the coset construction for kac moody algebras in terms of nonlinear integrable hierarchies subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of riemann surfaces to identify an action of thew algebras on the moduli space of stable curves and we give the invariants of this action this provides a generalization of the situation for the virasoro algebra where such as invariant is given by the so called mumford form which describes the partition function of the bosonic string

polynomial approximation local polynomial convexity and degenerate cr singularities abstract we begin with the following question given a closed disc d c and a complex valued function f c d is the uniform algebra on d generated by z and f equal to c d when f c 1 d this question is complicated by the presence of points in the surface s graph d f that have complex tangents such points are called cr singularities let p s be a cr singularity at which the order of contact of the tangent plane with s is greater than 2 i e a degenerate cr singularity we provide sufficient conditions for s to be locally polynomially convex at the degenerate singularity p this is useful because it is essential to know whether s is locally polynomially convex at a cr singularity in order to answer the initial question to this end we also present a general theorem on the uniform algebra generated by z and f which we use in our investigations this result may be of independent interest because it is applicable even to non smooth complex valued f

spin generalization of the calogero moser hierarchy and the matrix kp hierarchy we establish a correspondence between rational solutions to the matrix kp hierarchy and the spin generalization of the calogero moser system on the level of hierarchies namely it is shown that the rational solutions to the matrix kp hierarchy appear to be isomorphic to the spin calogero moser system in a sense that the dynamics of poles of solutions to the matrix kp hierarchy in the higher times is governed by the higher hamiltonians of the spin calogero moser integrable hierarchy with rational potential

infinite dimensional grassmannian structure of two dimensional quantum gravity we study the infinite dimensional grassmannian structure of 2d quantum gravity coupled to minimal conformal matters and show that there exists a large symmetry thew 1 symmetry using this symmetry structure we prove that the square root of the partition function which is a function of thep reduced kp hierarchy satisfies the vacuum condition of thew 1 algebra we further show that this condition is reduced to the vacuum condition of thew p algebra when the redundant variables for thep reduction are eliminated this mechanism also gives a prescription for extracting thew p algebra from thew 1 algebra

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

generalized orthogonal polynomials discrete kp and riemann hilbert problems classically a single weight on an interval of the real line leads to moments orthogonal polynomials and tridiagonal matrices appropriately deforming this weight with times t t 1 t 2 leads to the standard toda lattice and tau functions expressed as hermitian matrix integrals this paper is concerned with a sequence of t perturbed weights rather than one single weight this sequence leads to moments polynomials and a fuller matrix evolving according to the discrete kp hierarchy the associated tau functions have integral as well as vertex operator representations among the examples considered we mention nested calogero moser systems concatenated solitons and m periodic sequences of weights the latter lead to 2m 1 band matrices and generalized orthogonal polynomials also arising in the context of a riemann hilbert problem we show the riemann hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower times upper triangular matrix

rational solutions of the kp hierarchy and the dynamics of their poles ii construction of the degenerate polynomial solutions a general approach to constructing the polynomial solutions satisfying various reductions of the kadomtsev petviashvili kp hierarchy is described within this approach the reductions of the kp hierarchy are equivalent to certain differential equations imposed on the function of the hierarchy in particular the l reduction and the k constraint as well as their generalized counterparts are considered a general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the kdv and gardner equations the boussinesq and classical boussinesq systems the nls and yajima oikawa equations it is shown that the degenerate rational solutions of the kp hierarchy are related to stationary manifolds of the calogero moser cm hierarchy of dynamical systems the scattering dynamics of interacting particles in the cm systems may become complicated due to an anomalously slow fractional power rate of the particle motion along the stationary manifolds

rational ruijsenaars schneider hierarchy and bispectral difference operators abstract we show that a monic polynomial in a discrete variable n with coefficients depending on time variables t 1 t 2 is a function for the discrete kadomtsev petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of ruijsenaars schneider systems these functions were considered in l haine p iliev commutative rings of difference operators and an adelic flag manifold int math res not 2000 6 2000 281 323 where it was proved that they parametrize rank one solutions to a difference differential version of the bispectral problem

approximation by polynomials in two diffeomorphisms soit 1 k z et soient f et g des diffeomorphismes c k de c dans c ayant des degres opposes alors c f g est dense dans l espace de frechet c k c

connection between the existence of first integrals and the painleve property in two dimensional lotka volterra and quadratic systems taking advantage of the considerable amount of work done in the search for first integrals invariants for the two dimensional lotka volterra system and the quadratic system lvs and qs we compare the relations needed to exhibit invariants one for the lvs at least three for the qs to the two conditions of the painleve test index and compatibility we find that eventually restricting the invariants to those which are analytic all exponents integers and thereby adding new constraints these constraints always coalesce with the two painleve conditions we conclude that straightforward application of the painleve test picks up only these simple analytic invariants and that possession of the painleve property is too strong a condition for the existence of the invariants

kp hierarchy and trigonometric calogero moser hierarchy we consider trigonometric solutions of the kadomtsev petviashvili kp hierarchy it is known that their poles move as particles of the calogero moser model with trigonometric potential we show that this correspondence can be extended to the level of hierarchies the evolution of the poles with respect to the kth hierarchical time of the kp hierarchy is governed by a hamiltonian that is a linear combination of the first k higher hamiltonians of the trigonometric calogero moser hierarchy

the analogue of the backlund transformation for integrable many body systems canonical transformations analogous to the backlund transformation are discussed explicit formulae are given for many body systems of particles interacting in one dimension

an exact invariant for a class of time dependent anharmonic oscillators with cubic anharmonicity an exact invariant is constructed for a class of time dependent anharmonic oscillators using the method of the lie theory of extended groups the presence of the anharmonic term imposes a constraint on the nature of the time dependence for a subclass it is possible to obtain an energy like integral and a condition under which the motion is bounded

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

an explicit characterization of calogero moser systems combining theorems of halphen floquet and picard and a frobenius type analysis we characterize rational meromorphic simply periodic and elliptic kdv potentials in particular we explicitly describe the proper extension of the airault mckean moser locus associated with these three classes of algebro geometric solutions of the kdv hierarchy with special emphasis on the case of multiple collisions between the poles of solutions this solves a problem left open since the mid 1970s

time discretization of the spin calogero moser model and the semi discrete matrix kp hierarchy we introduce the discrete time version of the spin calogero moser system the equations of motion follow from the dynamics of poles of rational solutions to the matrix kadomtsev petviashvili hierarchy with discrete time the dynamics of poles is derived using the auxiliary linear problem for the discrete flow

bispectral kp solutions and linearization of calogero moser particle systems rational and soliton solutions of the kp hierarchy in the subgrassmanniangr1 are studied within the context of finite dimensional dual grassmannians in the rational case properties of the tau function which are equivalent to bispectrality of the associated wave function are identified in particular it is shown that there exists a bound on the degree of all time variables in if and only if is a rank one bispectral wave function the action of the bispectral involution in the generic rational case is determined explicitly in terms of dual grassmannian parameters using the correspondence between rational solutions and particle systems it is demonstrated that is a linearizing map of the calogero moser particle system and is essentially the map introduced by airault mckean and moser in 1977 2

classical w algebras we reconsider the relation between classicalw algebras and deformations of differential operators emphasizing the consistency with diffeomorphisms generators of thew algebra that arek differentials are constructed by a systematic procedure the method extends following drinfeld and sokolov tow algebras based on arbitrary simple lie algebras

solitons and almost intertwining matrices we define the algebraic variety of almost intertwining matrices to be the set of triples x y z of n n matrices for which xz yx t for a rank one matrix t a surprisingly simple formula is given for tau functions of the kp hierarchy in terms of such triples the tau functions produced in this way include the soliton and vanishing rational solutions the induced dynamics of the eigenvalues of the matrix x are considered leading in special cases to the ruijsenaars schneider particle system

symmetry singularities and integrability in complex dynamics ii rescaling and time translation in two dimensional systems the explicit integrability of second order ordinary differential equations invariant under time translation and rescaling is investigated quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the painleve and singularity properties of the different systems the transformation contains a parameter and for critical values intimately related to the possession of the painleve property in the parent second order equation one finds a difference from the generic behaviour this study is a prelude to a full discussion of the class of transformations which preserve the painleve property in the construction of quadratic systems from scalar nth order ordinary differential equations invariant under time invariant under time translation and rescaling

on a theorem of halphen and its application to integrable systems abstract we extend halphen s theorem which characterizes solutions of certain nth order differential equations with rational coefficients and meromorphic fundamental systems to a first order n n system of differential equations as an application of this circle of ideas we consider stationary rational algebro geometric solutions of the kdv hierarchy and illustrate some of the connections with completely integrable models of the calogero moser type in particular our treatment recovers the complete characterization of the isospectral class of such rational kdv solutions in terms of a precise description of the airault mckean moser locus of their poles

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

the local polynomial hull near a degenerate cr singularity bishop discs revisited let be a smooth real surface in and let be a point at which the tangent plane is a complex line how does one determine whether or not is locally polynomially convex at such a p i e at a cr singularity even when the order of contact of with at p equals 2 no clean characterisation exists difficulties are posed by parabolic points hence we study non parabolic cr singularities we show that the presence or absence of bishop discs around certain non parabolic cr singularities is completely determined by a maslov type index this result subsumes all known facts about bishop discs around order two non parabolic cr singularities sufficient conditions for bishop discs have earlier been investigated at cr singularities having high order of contact with these results relied upon a subharmonicity condition which fails in many simple cases hence we look beyond potential theory and refine certain ideas going back to bishop

a matrix integral solution to two dimensional w p gravity thep th gel fand dickey equation and the string equation l p 1 have a common solution expressible in terms of an integral overn n hermitean matrices for largen the integrand being a perturbation of a gaussian generalizing kontsevich s integral beyond the kdv case it is equivalent to showing that is a vacuum vector for aw generated from the coefficients of the vertex operator this connection is established via a quadratic identity involving the wave function and the vertex operator which is a disguised differential version of the fay identity the latter is also the key to the spectral theory for the two compatible symplectic structures of kdv in terms of the stress energy tensor associated with the virasoro algebra given a differential operator l d p q 2 t d p 2 cdots q p t with d frac partial dx t t 1 t 2 t 3 x equiv t 1 consider the deformation equations1 0 1 begin gathered frac partial l partial t n l n p l n 1 2 n 0 mod p hfill p reduced kp equation hfill end gathered ofl for which there exists a differential operatorp possibly of infinite order such that 0 2 l p 1 string equation in this note we give a complete solution to this problem in section 1 we give a brief survey of useful facts about thei function t the wave function t z solution of t n l n p x andl 1 p z and the corresponding infinitedimensional planev 0 of formal power series inz for largez v 0 span psi t z for all t in mathbb c infty in sato s grassmannian the three theorems below form the core of the paper their proof will be given in subseuqent sections each of which lives on its own right

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

local polynomial convexity of graphs of functions in several variables abstract in this paper we investigate the locally polynomial convexity of graphs of smooth functions in several variables we also give a sufficient condition for a real analytic function g defined near 0 in c which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on d for all disks d close enough to the origin in c

symmetries and first integrals for dissipative systems the connection between symmetries time independent and time dependent first integrals for autonomous systems of first order ordinary differential equations is discussed several new results which have been obtained by applying a reduce package are reported

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

explicit solution for p q duality in two dimensional quantum gravity by using sato s infinite dimensional grassmannian theory of the kp hierarchy we study the global structure of the theory space of 2d quantum gravity coupled to various minimal conformal fields labeled by a pair of integers p q after giving a rigorous proof of the equivalence of douglas s equation and the schwinger dyson equation w constrainton a function we establish thep q duality of the p q quantum gravity at green s function level as an application we discuss the metamorphosis of operators under unitarity preserving renormalization group flows