on a class of three dimensional integrable lagrangians we characterize non degenerate lagrangians of the form such that the corresponding euler lagrange equations are integrable by the method of hydrodynamic reductions the integrability conditions constitute an over determined system of fourth order pdes for the lagrangian density f which is in involution and possesses interesting differential geometric properties the moduli space of integrable lagrangians factorized by the action of a natural equivalence group is three dimensional familiar examples include the dispersionless kadomtsev petviashvili dkp and the boyer finley lagrangians f u x 3 3 u y 2 u x u t and respectively a complete description of integrable cubic and quartic lagrangians is obtained up to the equivalence transformations the list of integrable cubic lagrangians reduces to three examples there exists a unique integrable quartic lagrangian we conjecture that these examples exhaust the list of integrable polynomial lagrangians which are essentially three dimensional it was verified that there exist no polynomial integrable lagrangians of degree five we prove that the euler lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless lax pair

classification of integrable hydrodynamic chains and generating functions of conservation laws a new approach to classification of integrable hydrodynamic chains is established generating functions of conservation laws are classified by the method of hydrodynamic reductions the n parametric family of explicit hydrodynamic reductions allows us to reconstruct corresponding hydrodynamic chains many new hydrodynamic chains are found

on the fay identity for korteweg de vries tau functions and the identity for the wronskian of squared solutions of sturm liouville equation we show that the well known identity for the wronskian of squared solutions of a sturm liouville equation follows from the fay identity we also study some odd order 2n 1 order n 2 3 identities which are specific for tau functions related to the kdv hierarchy

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

characterizing jacobians via trisecants of the kummer variety we prove welters trisecant conjecture an indecomposable principally polarized abelian variety x is the jacobian of a curve if and only if there exists a trisecant of its kummer variety k x

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

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

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 integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

on fay identity in the first part of this paper we consider the transformation of the cubic identities for general korteweg de vries kdv tau functions from mishev j math phys 40 2419 2428 1999 to the specific identities for trigonometric kdv tau functions afterwards we consider the fay identity as a functional equation and provide a wide set of solutions of this equation the main result of this paper is theorem 3 4 where we generalize the identities from mishev an open problem is the transformation of the cubic identities from mishev to the specific identities for elliptic kdv tau functions

waterbag reductions of the dispersionless discrete kp hierarchy we consider the dispersionless limit of the discrete kp hierarchy we describe a family of n parameter waterbag reductions of the hierarchy and give sufficient conditions for the reduced equations of motion to be hyperbolic using geogdzhaev s method we solve the initial value problem for the reduced system the solution is implicit in generalized hodograph form tsarev s p 1985 sov math dokl 31 488 the relationship of this form of the solution to the properties of the hierarchy the reduction and the initial data is discussed briefly indicating how this construction might be generalized

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

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

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

enumerative geometry tau functions and heisenberg virasoro algebra in this paper we establish relations between three enumerative geometry tau functions namely the kontsevich witten hurwitz and hodge tau functions the relations allow us to describe the tau functions in terms of matrix integrals virasoro constraints and kac schwarz operators all constructed operators belong to the algebra or group of symmetries of the kp hierarchy

multiscale autoregressive models and wavelets the multiscale autoregressive mar framework was introduced to support the development of optimal multiscale statistical signal processing its power resides in the fast and flexible algorithms to which it leads while the mar framework was originally motivated by wavelets the link between these two worlds has been previously established only in the simple case of the haar wavelet the first contribution of this paper is to provide a unification of the mar framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem the second contribution of this paper is to develop wavelet based approximate internal mar models for stochastic processes this will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients taking advantage of the statistical machinery provided by the mar framework we will illustrate the application of our models to sample path generation and estimation from noisy irregular and sparse measurements

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

integrability of hurwitz partition functions partition functions often become functions of integrable hierarchies if they are considered dependent on infinite sets of parameters called time variables the hurwitz partition functions z rd2 kr r t 1 r t k exp n ncr n depend on two types of such time variables t and kp toda integrability in t requires that k 2 and also that cr n are selected in a rather special way in particular the naive cut and join operators are not allowed for n 2 integrability in further restricts the choice of cr n forbidding for example the free cumulants it also requires that k 1 the quasi classical integrability the wdvv equations is naturally present in variables but also requires a careful definition of the generating function

differential geometric approach to the integrability of hydrodynamic chains the haantjes tensor the integrability of an m component system of hydrodynamic type u t v u u x by the generalized hodograph method requires the diagonalizability of the m m matrix v u this condition is known to be equivalent to the vanishing of the corresponding haantjes tensor we generalize this approach to hydrodynamic chains infinite component systems of hydrodynamic type for which the matrix v u is sufficiently sparse for such systems the haantjes tensor is well defined and the calculation of its components involves finite summations only we illustrate our approach by classifying broad classes of conservative and hamiltonian hydrodynamic chains with the zero haantjes tensor we prove that the vanishing of the haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi hamiltonian hydrodynamic reductions thus providing an easy to verify necessary condition for the integrability

p adic theta functions and solutions of the kp hierarchy based on schottky uniformization theory of riemann surfaces we construct a universal power series for riemann theta function solutions of the kp hierarchy specializing this power series to the coordinates associated with schottky groups overp adic fields we show that thep adic theta functions of mumford curves give solutions of the kp hierarchy

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

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

a criterion for jacobi varieties is a 2 1 map onto its image the kummer wirtinger variety kw x associated with x the following is well known cf 6 cf also 10 assume that x is the polarized jacobian of a smooth curve c and consider c as embedded in x by an arbitrary element of picl c let d be an effective divisor of degree 3 on c and call y the subscheme of x defined by d put s s y the abelian sum of the components of the corresponding 0 cycle of degree 3 on x then for any d ex with 2d e c s there exists a line 1 in pn such that the translated subscheme d y satisfies d y c 4 1 l in other words 1 is a trisecant or a flex of the kummer wirtinger variety and the image of d y lies in the intersection of 1 with kw x in his paper 3 r c gunning has shown how to use the above property in order to characterize jacobians gunning s result can be stated as follows

cohomological structure in soliton equations and jacobian varieties on etudie la structure des orbites du systeme dynamique definie par la hierarchie kp totale on montre que chaque orbite est localement isomorphe a un certain groupe de cohomologie associe a une algebre commutative le systeme dynamique kp restreint a chaque orbite determine un systeme dynamique de mouvements lineaires sur lui par rapport a la structure lineaire du groupe de cohomologie

a lax representation for the vertex operator and the central extension integrable hierarchies viewed as isospectral deformations of an operatorl may admit symmetries they are time dependent vector fields transversal to and commuting with the hierarchy and forming an algebra in this work the commutation relations for the symmetries are shown to be based on a non commutative lie algebra splitting theorem the symmetries viewed as vector fields onl are expressed in terms of a lax pair

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

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

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

identification of nonlinear spatiotemporal dynamical systems with nonuniform observations using reproducing kernel based integral least square regulation the identification of nonlinear spatiotemporal dynamical systems given by partial differential equations has attracted a lot of attention in the past decades several methods such as searching principle based algorithms partially linear kernel methods and coupled lattice methods have been developed to address the identification problems however most existing methods have some restrictions on sampling processes in that the sampling intervals should usually be very small and uniformly distributed in spatiotemporal domains these are actually not applicable for some practical applications in this paper to tackle this issue a novel kernel based learning algorithm named integral least square regularization regression ilsrr is proposed which can be used to effectively achieve accurate derivative estimation for nonlinear functions in the time domain with this technique a discretization method named inverse meshless collocation is then developed to realize the dimensional reduction of the system to be identified thereafter with this novel inverse meshless collocation model the ilsrr and a multiple kernel based learning algorithm a multistep identification method is systematically proposed to address the identification problem of spatiotemporal systems with pointwise nonuniform observations numerical studies for benchmark systems with necessary discussions are presented to illustrate the effectiveness and the advantages of the proposed method

on local description of two dimensional geodesic flows with a polynomial first integral in this paper we present a construction of multiparametric families of two dimensional metrics with a polynomial first integral of arbitrary degree in momenta such integrable geodesic flows are described by solutions of some semi hamiltonian hydrodynamic type system we give a constructive algorithm for the solution of the derived hydrodynamic type system i e we found infinitely many conservation laws and commuting flows thus we were able to find infinitely many particular solutions of this hydrodynamic type system by the generalized hodograph method therefore infinitely many particular two dimensional metrics equipped with first integrals polynomial in momenta were constructed

periodic multiphase solutions of the kadomtsev petviashvili equation n phase solutions of the kadomtsev petviashvili kp equation 1970 that are periodic in space variables x and y were obtained and effectively investigated using the schottky uniformisation of which a short description is given many wave patterns are represented graphically as contour plots and as isometric projections for different parameter values of two three and four phase solutions of the kp equation

identification of continuous spatiotemporal systems we present a method for the identification of continuous spatiotemporal dynamics from experimental data we use a model in the form of a partial differential equation and formulate an optimization problem for its estimation from data the solution is found as a multivariate nonlinear regression problem using the ace algorithm the procedure is successfully applied to data obtained by simulation of the swift hohenberg equation there are no restrictions on the dimensionality of the investigated system allowing for the analysis of high dimensional chaotic as well as transient dynamics the demands on the experimental data are discussed as well as the sensitivity of the method towards noise

classification of integrable two component hamiltonian systems of hydrodynamic type in 2 1 dimensions hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics the whitham averaging procedure and the theory of frobenius manifolds in 1 1 dimensions the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable hamiltonians depend on a certain number of arbitrary functions of two variables on the contrary in 2 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions which is a natural analogue of the generalised hodograph transform in higher dimensions leads to finite dimensional moduli spaces of integrable hamiltonians in this paper we classify integrable two component hamiltonian systems of hydrodynamic type for all existing classes of differential geometric poisson brackets in 2d establishing a parametrisation of integrable hamiltonians via elliptic hypergeometric functions our approach is based on the godunov type representation of hamiltonian systems and u

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

the riemann liouville integral and continuous kp hierarchy extension of the standard construction of the kadomtsev petviashvili kp hierarchy by the use of riemann liouville integral is given in consequence we obtain the new classes of integer as well as fractional graded kp hierarchies which are further investigated the fractional calculus leads to the new generalization of the w algebra

supertori are algebraic curves super riemann surfaces of genus 1 with arbitrary spin structures are shown to be the sets of zeroes of certain polynomial equations in projective superspace we conjecture that the same is true for arbitrary genus properties of superelliptic functions and super theta functions are discussed the boundary of the genus 1 super moduli space is determined

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

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

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

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 kp equation with quasiperiodic initial data abstract the kadomtsev petviashvili kp equation is known to admit exact quasiperiodic solutions that can be written in terms of riemann theta functions with a finite number of phases in each solution in this paper we propose a method to solve the initial value problem for the kp equation for initial data taken from this class of quasiperiodic functions

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

identification of partial differential equation models for a class of multiscale spatio temporal dynamical systems in this article the identification of a class of multiscale spatio temporal dynamical systems which incorporate multiple spatial scales from observations is studied the proposed approach is a combination of adams integration and an orthogonal least squares algorithm in which the multiscale operators are expanded using polynomials as basis functions and the spatial derivatives are estimated by finite difference methods the coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a b spline wavelet multi resolution analysis the resulting identified models of the spatio temporal evolution form a system of partial differential equations with different spatial scales examples are provided to demonstrate the efficiency of the proposed method

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

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

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

in search of periodic solutions for a reduction of the benney chain we search for smooth periodic solutions for the system of quasi linear equations known as the lax dispersionless reduction of the benney moments chain it is naturally related to the existence of a polynomial in momenta integral for a classical hamiltonian system with 1 5 degrees of freedom for the solution in question it is not known a priori if the system is elliptic or hyperbolic or of mixed type we consider two possible regimes for the solution the first is the case of only one real eigenvalue where we can completely classify the solutions the second case of strict hyperbolicity is really a challenge we find a remarkable 2 2 reduction which is strictly hyperbolic with one umbilic point but violates the condition of genuine non linearity

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

the kadomtsev petviashvili equations and fundamental string theory in this paper the infinite sequence of non linear partial differential equations known as the kadomtsev petviashvili equations is described in simple terms and possible applications to a fundamental description of interacting strings are addressed lines of research likely to prove useful in formulating a description of non perturbative string configurations are indicated

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

algebro geometric approach in the theory of integrable hydrodynamic type systems the algebro geometric approach for integrability of semi hamiltonian hydrodynamic type systems is presented the class of symmetric hydrodynamic type systems is defined and the calculation of the associated riemann surfaces is greatly simplified for this class many interesting and physically motivated examples are investigated

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

functional representation of the ablowitz ladik hierarchy the ablowitz ladik hierarchy alh is considered in the framework of the inverse scattering approach after establishing the structure of solutions of the auxiliary linear problems the alh which has been originally introduced as an infinite system of differential difference equations is presented as a finite system of difference functional equations the representation obtained when rewritten in terms of hirota s bilinear formalism is used to demonstrate relations between the alh and some other integrable systems the kadomtsev petviashvili hierarchy in particular

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

the geometry of the super kp flows a supersymmetric generalization of the krichever map is used to construct algebro geometric solutions to the various super kadomtsev petviashvili skp hierarchies the geometric data required consist of a suitable algebraic supercurve of genusg generallynot a super riemann surface with a distinguished point and local coordinates z there and a generic line bundle of degreeg 1 with a local trivialization near the point the resulting solutions to the manin radul skp system describe coupled deformations of the line bundle and the supercurve itself in contrast to the ordinary kp system which deforms line bundles but not curves two new skp systems are introduced an integrable jacobian system whose solutions describe genuine jacobian flows deforming the bundle but not the curve and a nonintegrable maximal system describing independent deformations of bundle and curve the kac van de leur skp system describes the same deformations as the maximal system but in a different parametrization

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

an inflectionary tangent to the kummer variety and the jacobian condition in this paper we deal with a degenerate version of the trisecant conjecture let x theta be an indecomposable principally polarized abelian variety and let theta be a symmetric representative of the polarization we shall denote by theta a non zero section of the corresponding sheaf the linear system vert 2 theta vert is base point free and it is independent of the choice of theta the image of the morphism k x rightarrow vert 2 theta vert associated with the base point free linear system vert 2 theta vert is a projective variety which is called the kummer variety of x theta welters conjectured that the existence of one trisecant line to the kummer variety characterizes the jacobians it is well known that the kummer variety of a jacobian has a rich geometry in terms of trisecants and flexes we prove that if there exists an inflectionary tangent l to the kummer variety associated with x theta then x theta is a jacobian provided that there are no set theoretical d invariant components of the scheme d theta theta cap d theta 0 where d is an invariant vector field on x associated to l

the characterization of two component 2 1 dimensional integrable systems of hydrodynamic type we obtain the necessary and sufficient conditions for a two component 2 1 dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions these conditions are in involution implying that the systems in question are locally parametrized by 15 arbitrary constants it is proved that all such systems possess three conservation laws of hydrodynamic type and therefore are symmetrizable in godunov s sense moreover all such systems are proved to possess a scalar pseudopotential which plays the role of the dispersionless lax pair we demonstrate that the class of two component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions thus establishing the equivalence of the two possible definitions of integrability explicit linearly degenerate examples are constructed

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

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

identification of finite dimensional models of infinite dimensional dynamical systems the identification of finite dimensional discrete time models of deterministic linear and nonlinear infinite dimensional systems from pointwise observations is investigated the input and output observations are used to construct finite dimensional approximations of the solution and the forcing function which are expanded in terms of a finite element basis an algorithm to determine a minimal basis to approximate the data is introduced subsequently the resulting coordinate vectors are used to identify a finite dimensional discrete time model theoretical results concerning the existence stability and convergence of the finite dimensional representation are established numerical results involving identification of finite dimensional models for both linear and nonlinear infinite dimensional systems are presented

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

equations of the moduli of pointed curves in the infinite grassmannian the main result of this paper is the explicit computa tion of the equations defining the moduli space of triples c p where c is an integral and complete algebraic curve p a smooth rational point anda certain isomorphism this is achieved by introducing algebraically infinite grassmannians tau and baker ahkiezer functions and by proving an addition formula for tau functions