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

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

a new look at second order equations and lagrangian mechanics to each second order equation field gamma on a tangent bundle tm the authors associate a particular subset xgamma of 1 forms on tm lagrangian systems then are characterised by the property that xgamma contains an element which is non degenerated and exact for general second order equation fields gamma they study type 1 1 tensor fields preserving both xgamma and a kind of dual set of vector fields xgamma finally they establish some theorems concerning xgamma which cover known results in lagrangian mechanics

upper bounds for the distances between consecutive zeros of solutions of first order delay differential equations abstract by introducing a new class of sequences involving iterates of the delay we are able to derive a sharper estimate on the ratio x t x t based on this we get some new upper bounds for the distance between consecutive zeros of solutions of the first order delay differential equation x t p t x t 0 in particular our results are not covered by previously known results some examples and a table are given to illustrate our results

constructing multiply connected quadrature domains multiply connected bounded quadrature domains with finite connectivity are re constructed from their quadrature data using conformal mappings that are ratios of products of schottky klein prime functions this method provides the natural generalization of the conformal maps to simply and doubly connected quadrature domains constructed by the first author in a num ber of physical applications the efficacy of the method is demonstrated by the explicit construction of a range of examples as well as by comparison with alternative constructive methods recently in troduced by crowdy r soc lond proc ser a math phys eng sci 457 2001 pp 2337 2359 and richardson european j appl math 12 2001 pp 571 599

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

on a class of geometry driven free boundary problems an abstract class of two dimensional geometry driven free boundary problems is considered in which the evolution of the cauchy transform of the time evolving domain satisfies a partial differential equation of conservation type this idea generalizes an association between free boundary problems and the inverse gravitational problem it is shown that this class of free boundary problems admits exact solutions expressible in terms of a finite set of time evolving parameters in addition even when exact solutions are not available a theorem is established which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities two distinct problems of physical interest are shown to fall within this abstract class the evolution of a fluid blob in a rotating hele shaw cell and the evolution of highly viscous fluid under the effects of surface tension various aspects of these two problems are discussed

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

uniformizing the boundaries of multiply connected quadrature domains using fuchsian groups abstract this paper proposes a new method for uniformizing the boundaries of multiply connected quadrature domains by means of conformal maps that are automorphic functions invariant under the action of a schottky group that is a fuchsian group of the second kind the conformal maps are expressed as ratios of schottky klein prime functions the result is significant because a classical infinite product formula for the schottky klein prime function associated with fuchsian groups of the second kind is known to be convergent

symmetries first integrals and the inverse problem of lagrangian mechanics deals with the following question given a symmetry vector field y of a system of second order ordinary differential equations and an associated constant of the motion f is it possible to find a lagrangian l for the system such that y becomes a noether symmetry with respect to l and f its implied noether constant for one degree of freedom systems the answer to this question is affirmative in addition attention is paid to the construction of a suitable constant of the motion f for given symmetry y and vice versa several examples are discussed

symmetry theory and lagrangian inverse problem for time dependent second order differential equations a set xgamma of vector fields in the evolution space e playing the role of newtonian vector fields with respect to a second order equation field gamma is introduced and endowed with a cinfinity e module structure a dual set m gamma is used for giving an answer to the lagrangian inverse problem the symmetry theory is also developed in this framework and in particular the characterisation of symmetries of gamma in terms of the transformation properties of the lagrangian l is also given

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

multipolar vortices and algebraic curves this paper demonstrates that analytical solutions to the steady two dimensional euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains the solutions consist of a finite set of line vortices superposed on finite area patches of uniform vorticity by way of example new solutions are presented in which the support of the vorticity is a quintuply connected vortex patch with five vorticity maxima modelling a symmetric pentapolar vortex consisting of a central core vortex four satellite vortices and four enclosed zones of irrotational fluid between the core and the satellite vortices the method of construction is amenable to the derivation of exact solutions of the two dimensional euler equations having vorticity distributions of even greater geometrical complexity

a necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments abstract in this article we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions in particular we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non negative examples and tables are given to illustrate our results

vortex patch equilibria of the euler equation and random normal matrices a new analogy is drawn between vortex patch or v state equilibria of the euler equations for ideal fluids and the planar limit of random normal matrix models physically the former is a quite different fluid dynamical problem to hele shaw flow or laplacian growth to which an analogy with matrix models has become well known in recent years the connection of random matrices with vortex dynamics is made via the so called modified schwarz potential this theoretical link while interesting in itself has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics hence for multi support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a schottky model of the underlying algebraic curves and use of the schottky klein prime function two support eigenvalue distributions in a quartic potential are given as an illustrative example

regular articlethe distribution of zeros of solutions of differential equations with a variable delay in this paper the distribution of zeros of solutions of the first order differential equation with a variable delayx t ptx t 0is studied the estimate for the distance between adjacent zeros of the oscillatory solution of the above equation is obtained

on the distance between consecutive zeros of solutions of first order delay differential equations this paper is concerned with the distribution of zeros of solutions of first order linear delay differential equations with variable coefficients of the formx t p t x email protected 0 t t where t 0 p t c t 0 by introducing a class of polynomial functions we are able to derive new estimates for the lower and upper bounds of the distance between consecutive zeros of solutions of the above equations we illustrate the obtained results with several examples

on the distribution of zeros of solutions of first order delay differential equations abstract this paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation x t p t x t 0 where p is a nonnegative continuous function our results are new and improve some known criteria illustrative examples are given to show the strength of some obtained results over the known ones

adjoint symmetries separability and volume forms two results of a preceding paper are generalized the first is about characterizing to what extent preservation of the energy function of a lagrangian of mechanical type turns dynamical symmetries into noether symmetries the generalization here is twofold polynomial integrals of arbitrary degree are considered and the kinetic energy can have an arbitrary metric the second result here again for arbitrary metrics is about the way separation variables for the hamilton jacobi equation when they are ensured to exist by eisenhart s theorem can be computed in principle from a factorization property of a certain volume form the main novelty in the way the generalizations are discussed is that the emphasis is shifted from symmetries to the dual concept of adjoint symmetries

conformal mappings and dispersionless toda hierarchy ii general string equations in this article we classify the solutions of the dispersionless toda hierarchy into degenerate and non degenerate cases we show that every non degenerate solution is determined by a function mathcal h z 1 z 2 of two variables we interpret these non degenerate solutions as defining evolutions on the space mathfrak d of pairs of conformal mappings g f where g is a univalent function on the exterior of the unit disc f is a univalent function on the unit disc normalized such that g f 0 0 and f 0 g 1 for each solution we show how to define the natural time variables t n n in mathbb z as complex coordinates on the space mathfrak d we also find explicit formulas for the tau function of the dispersionless toda hierarchy in terms of mathcal h z 1 z 2 imposing some conditions on the function mathcal h z 1 z 2 we show that the dispersionless toda flows can be naturally restricted to the subspace of mathfrak d defined by f w 1 overline g 1 bar w this recovers the result of zabrodin theor math phy 12 1511 1525 2001

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

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

integrable structure of the dirichlet boundary problem in two dimensions we study how the solution of the two dimensional dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem we show that the hadamard formula for the variation of the dirichlet green function under deformations of the domain reveals an integrable structure the independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain the solution to the dirichlet boundary problem is expressed through the tau function of the dispersionless toda hierarchy we also discuss a degenerate case of the dirichlet problem on the plane with a gap in this case the tau function is identical to the partition function of the planar large n limit of the hermitian one matrix model

martingales of stochastic laplacian growth abstract a family of exponential martingales of a stochastic laplacian growth problem is proposed stochastic laplacian growth describes a regularized interface dynamics in a two fluid system where the viscous fluid is incompressible at a large scale while compressible at a small scale in the vicinity of the interface hence random fluctuations of pressure near the boundary are inevitable by using loewner kufarev equation we study interface dynamics generated by nonlocal random loewner measure which produces the patterns with viscous fingers we use a schottky double construction to introduce a one parametric family of functions of random processes on the double closely connected to the correlation functions of primary operators of the boundary conformal field theory in the coulomb gas framework for a specific value of the parameter these functions are martingales with respect to stochastic loewner flow on the schottky double a connection between the proposed algebraic construction and the physical problem of stochastic interface dynamics in the hele shaw cell relies on the hadamard s variational formula namely the variation of pressure in stochastic laplacian growth in the vicinity of the interface is given by the variance of martingales on the double

adjoint symmetries for time dependent second order equations the authors extend part of their previous work on autonomous second order systems sarlet et al 1987 to time dependent differential equations the main subject of the paper concerns the notion of adjoint symmetries they are introduced as a particular type of 1 form whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields it is shown that all interesting properties of adjoint symmetries known from the autonomous theory have their counterparts in the present framework of particular interest is a result establishing that lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries a number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way

the distribution of zeros of solutions of first order delay differential equations abstract we give two estimates of the distance between adjacent zeros of the solutions of the first order delay differential equation x t p t x t 0

shocks and finite time singularities in hele shaw flow abstract hele shaw flow at vanishing surface tension is ill defined in finite time the flow develops cusp like singularities we show that this ill defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid the graph of shocks grows and branches velocity and pressure have finite discontinuities across the shock we formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid we also formulate the dispersive weak solution in algebro geometrical terms as an evolution of the krichever boutroux complex curve we study in detail the most generic 2 3 cusp singularity which gives rise to an elementary branching event this solution is self similar and expressed in terms of elliptic functions

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

the symmetry in the structure of dynamical and adjoint symmetries of second order differential equations with each second order differential equation z in the evolution space j1 mn 1 we associate using a new differential operator az four families of vector fields and 1 forms on j1 mn 1 providing a natural set up for the study of symmetries first integrals and the inverse problem for z we analyse the relations between the four families pointing out the symmetric structure of this set up when a lagrangian for z exists the bijection between dynamical and dual symmetries is included in the whole context suggesting the corresponding bijection between dual adjoint and adjoint symmetries as an application we show how some results of the inverse problem can be framed naturally in this geometrical context

a geometric approach to constrained mechanical systems symmetries and inverse problems we introduce a geometrical framework for the description of constrained mechanical systems and we analyse different kinds of symmetries and their relationships we propose a new definition for non holonomic lagrangian mechanical systems and we give a geometrical characterization for the helmholtz conditions related to the inverse problem

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

multi cut solutions of laplacian growth a new class of solutions to laplacian growth lg with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases these solutions which are time dependent conformal maps with branch cuts inside the unit circle are governed by a nonlinear integral equation and describe oil fjords with non parallel walls in viscous fingering experiments in hele shaw cells integrals of motion for the multi cut lg solutions in terms of singularities of the schwarz function are found and the dynamics of densities jumps on the cuts are derived the subclass of these solutions with linear cauchy densities on the cuts of the schwarz function is of particular interest because in this case the integral equation for the conformal map becomes linear these solutions can also be of physical importance by representing oil air interfaces which form oil fjords with a constant opening angle in accordance with recent experiments in a hele shaw cell

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

pseudo symmetries noether s theorem and the adjoint equation pseudo symmetries were introduced by sarlet and cantrijn 1984 for time dependent non conservative systems they are reconsidered in the context of general autonomous second order systems relying on the new approach to such systems which was presented by sarlet et al they further introduce the notion of adjoint symmetries of a second order system as being associated to invariant 1 forms and show how they may be related to first integrals or to lagrangians under appropriate circumstances their results enable them to clarify a rather unusual account of noether s theorem which was recently given by gordon 1986

the distribution of zeros of solutions of neutral differential equations in this paper the distribution of zeros of solutions of the first order neutral differential equation with two variable delays x t p t x g t q t x h t 0is studied the estimate for the distance between adjacent zeros of oscillatory solutions of the above equation is obtained

bubble break off in hele shaw flows singularities and integrable structures abstract bubbles of inviscid fluid surrounded by a viscous fluid in a hele shaw cell can merge and break off during the process of break off a thinning neck pinches off to a universal self similar singularity we describe this process and reveal its integrable structure it is a solution of the dispersionless limit of the akns hierarchy the singular break off patterns are universal not sensitive to details of the process and can be seen experimentally we briefly discuss the dispersive regularization of the hele shaw problem and the emergence of the painleve ii equation at the break off

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

oscillation of first order delay equations we study the oscillation of first order delay equations using a method that parallels the use of riccati equations in the study of the oscillation of second order ordinary differential equations without delay we give an alternative proof of a comparison theorem first established by kwong and patula obtain some asymptotic behaviors of nonoscillatory solutions prove a new criterion that falls within a gap left open by established results and confirm a conjecture raised by hunt and yorke