on the jacobi last multiplier integrating factors and the lagrangian formulation of differential equations of the painleve gambier classification abstract we use a formula derived almost seventy years ago by madhav rao connecting the jacobi last multiplier of a second order ordinary differential equation and its lagrangian and determine the lagrangians of the painleve equations indeed this method yields the lagrangians of many of the equations of the painleve gambier classification using the standard legendre transformation we deduce the corresponding hamiltonian functions while such hamiltonians are generally of non standard form they are found to be constants of motion on the other hand for second order equations of the lienard class we employ a novel transformation to deduce their corresponding lagrangians we illustrate some particular cases and determine the conserved quantity first integral resulting from the associated noetherian symmetry finally we consider a few systems of second order ordinary differential equations and deduce their lagrangians by exploiting again the relation between the jacobi last multiplier and the lagrangian

gas flows with several thermal nonequilibrium modes we study gas flows with any finite number of thermal nonequilibrium modes the equations describing such flows are a hyperbolic system with several relaxation equations an important feature is entropy increase dictated by physics for any irreversible process under physical assumptions we obtain properties of thermodynamic variables relevant to stability by the energy method we prove global existence and uniqueness for the cauchy problem when the initial data are small perturbations of constant equilibrium states we give a precise formulation of the fundamental solution for the linearized system and use it to obtain large time behavior of solutions to the nonlinear system in particular we show that the entropy increases but stays bounded the resulting asymptotic picture of nonequilibrium flows is in a pointwise sense both in space and in time

singular perturbations of first order hyperbolic systems with stiff source terms abstract this work develops a singular perturbation theory for initial value problems of nonlinear first order hyperbolic systems with stiff source terms in several space variables it is observed that under reasonable assumptions many equations of classical physics of that type admit a structural stability condition this condition is equivalent to the well known subcharacteristic condition for one dimensional 2 2 systems and the well known time like condition for one dimensional scalar second order hyperbolic equations with a small positive parameter multiplying the highest derivatives under this stability condition we construct formal asymptotic approximations of the initial layer solution to the nonlinear problem furthermore assuming some regularity of the solutions to the limiting inner problem and the reduced problem we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval the stability condition seems to be a key to problems of this kind and can be easily verified moreover this presentation unifies and improves earlier works for some specific equations

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

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

decay of the navier stokes poisson equations we establish the time decay rates of the solution to the cauchy problem for the compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates as a corollary we also obtain the usual lp l2 1 p 2 type of the optimal decay rates compared to the compressible navier stokes system and the compressible irrotational euler poisson system our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

discretising systematically the painleve equations we apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the painleve equations the important point is that in the case of the painleve equations one needs to preserve the integrability we show that this is indeed possible using our systematic procedure applied to the equations obtained from the hamiltonian description of the painleve equations

thermal non equilibrium flows in three space dimensions we study the equations describing the motion of a thermal non equilibrium gas in three space dimensions it is a hyperbolic system of six equations with a relaxation term the dissipation mechanism induced by the relaxation is weak in the sense that the shizuta kawashima criterion is violated this implies that a perturbation of a constant equilibrium state consists of two parts one decays in time while the other stays in fact the entropy wave grows weakly along the particle path as the process is irreversible we study thermal properties related to the well posedness of the nonlinear system we also obtain a detailed pointwise estimate on the green s function for the cauchy problem when the system is linearized around an equilibrium constant state the green s function provides a complete picture of the wave pattern with an exact and explicit leading term comparing with existing results for one dimensional flows our results reveal a new feature of three dimensional flows not only does the entropy wave not decay but the velocity also contains a non decaying part strongly coupled with its decaying one the new feature is supported by the second order approximation via the chapman enskog expansions which are the navier stokes equations with vanished shear viscosity and heat conductivity

generalized huygens principle for bipolar non isentropic compressible navier stokes poisson system in dimension three abstract the generalized huygens principle for the non isentropic bipolar navier stokes poisson system in dimension three is given which is different from the unipolar case all of the previous estimates containing huygens waves on wave behaviors for compressible flow models rely heavily on the conservative structure for instance the navier stokes system and even the isentropic bipolar navier stokes poisson system the reason is that the huygens waves decay slower than diffusion waves in l p r 3 space when 1 p 2 however the non isentropic case here is not conservative due to two energy equations and moreover it is not yet conservative even if one reformulates the original system as the isentropic case in 26 the main contribution is that we first solve the nonlinear problem when there exists the huygens wave in the green s function but the nonlinear system is not conservative which is achieved by establishing refined pointwise estimates for the green s function and nonlinear convolution estimates

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

an affine weyl group approach to the eight parameter discrete painleve equation we present a geometrical construction of the eight parameter discrete painleve equations our starting point is the e 1 8 affine weyl group we assume that the multi dimensional function lives on the vertices of the weight lattice of this group we derive the bilinear equations related to the discrete painleve equation in the form of nonautonomous hirota miwa equations and the elementary miura transformations the compatibility condition of the various miura s that can be written leads to three types of equations difference multiplicative q and another type where the parameters and the independent variable enter through the arguments of elliptic functions we write explicitly the discrete equations in the first two cases and produce their degeneration through coalescence of parameters

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

concernant la relation de distribution satisfaite par la fonction φ associee a un reseau complexe for any pair of latticesl andl satisfying i l l and ii the indexn ofl intol is prime to 6 we construct from the usual function ofl cf no 1 some elliptic function psi psi z l underset raise0 3em hbox smash scriptscriptstyle l of the variablez with period latticel and divisor n 0 l sum limits i 1 n t i l over the torus l where the complex numberst i 1 i n describe a complete set of representatives of the quotientl l

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

l1 asymptotic behavior of compressible isentropic viscous 1 d flow we study the large time behavior in l1 of the compressible isentropic viscous 1 d flow under the assumption that the initial data are smooth and small we show that the solutions are approximated by the solutions of a parabolic system and in turn by diffusion waves which are solutions of burgers equations decay rates in l1 are obtained our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system 1994 john wiley sons inc

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation we study gas flow in vibrational nonequilibrium the model is a 4 4 nonlinear hyperbolic system with relaxation under physical assumptions properties of thermodynamic variables relevant to stability are obtained global existence for cauchy problems with smooth and small data is established and large time behavior is studied in the pointwise sense we formulate the fundamental solution in a systematic way for a general linear system with relaxation the fundamental solution provides insights to the behavior of the nonlinear system and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow we also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic parabolic systems and has now proved to be important also for hyperbolic systems with relaxation

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

solving the non isentropic navier stokes equations in odd space dimensions the green function method in this paper we investigate the wave structure of the solution for the compressible navier stokes equations in both one space dimension and three space dimensions first we give the pointwise estimate on the green function for the linearized system by dividing the physical domain into two parts which are the finite mach number region and outside finite mach number region we use long wave short wave decomposition and a weighted energy estimate method in each region separately the green function provides a complete picture of the wave pattern with explicit leading terms then with the help of the duhamel principle we give the pointwise estimate of the solution for the nonlinear problems 1 1 and 1 2 specially we find that the decay rate in lp r3 2 p for the energy is faster than those for the density and the momentum m

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

pointwise estimates of solution for the navier stokes poisson equations in multi dimensions abstract the cauchy problem of the navier stokes poisson equations in multi dimensions n 3 is considered we obtain the pointwise estimates of the solution when it is a perturbation of the constant state then we have the optimal l p r n p 1 convergence rate of the solution

kernels and point processes associated with whittaker functions this article considers whittaker s confluent hypergeometric function w where is real and is real or purely imaginary then x x 1 2w x arises as the scattering function of a continuous time linear system with state space l2 1 2 and input and output spaces c the hankel operator on l2 0 is expressed as a matrix with respect to the laguerre basis and gives the hankel matrix of moments of a jacobi weight w0 x xb 1 x a the operation of translating is equivalent to deforming w0 to give wt x e t xxb 1 x a the determinant of the hankel matrix of moments of we satisfies the form of painleve s transcendental differential equation pv it is shown that gives rise to the whittaker kernel from random matrix theory as studied by borodin and olshanski commun math phys 211 335 358 2000 whittaker kernels are closely related to systems of orthogonal polynomials for a pollaczek jacobi type weight lying outside the usual szego class

rational solutions of the discrete time toda lattice and the alternate discrete painleve ii equation the yablonskii vorob ev polynomials yn t which are defined by a second order bilinear differential difference equation provide rational solutions of the toda lattice they are also polynomial tau functions for the rational solutions of the second painleve equation pii here we define two variable polynomials yn t h on a lattice with spacing h by considering rational solutions of the discrete time toda lattice as introduced by suris these polynomials are shown to have many properties that are analogous to those of the yablonskii vorob ev polynomials to which they reduce when h 0 they also provide rational solutions for a particular discretization of pii namely the so called alternate discrete pii and this connection leads to an expression in terms of the umemura polynomials for the third painleve equation piii it is shown that the backlund transformation for the alternate discrete painleve equation is a symplectic map and the shift in time is also symplectic finally we present a lax pair for the alternate discrete pii which recovers jimbo and miwa s lax pair for pii in the continuum limit h 0

the decay rate of solution for the bipolar navier stokes poisson system in this paper the cauchy problem for the bipolar navier stokes poisson system is studied we obtain the global existence and the hs decay rate of classical solutions for this problem our approach is mainly based on the analysis of the green s function of the linearized system and some elaborate energy estimates it should be mentioned that with the help of long wave short wave decomposition the decay rate of the higher order derivatives of the solutions is obtained furthermore based on the hs decay rate of the solutions we also give the lp estimate of the solution

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

the discrete painleve ii equations miura and auto backlund transformations we present miura transformations for all discrete painleve ii equations known to date we then use these miuras to derive special solutions in terms of discrete airy functions and to construct auto backlund transformations for the discrete painleve equations these transformations are then used to generate rational solutions some new forms of d pii and d p34 are obtained as well

computing rational points on rank 1 elliptic curves via l series and canonical heights let e q be an elliptic curve of rank 1 we describe an algorithm which uses the value of l e 1 and the theory of canonical heghts to efficiently search for points in e q and e z s for rank 1 elliptic curves e q of moderately large conductor say on the order of 10 7 to 10 10 and with a generator having moderately large canonical height say between 13 and 50 our algorithm is the first practical general purpose method for determining if the set e z s contains non torsion points

painleve vi and hankel determinants for the generalized jacobi weight we study the hankel determinant of the generalized jacobi weight x t x 1 x for x 0 1 with 0 t 0 and based on the ladder operators for the corresponding monic orthogonal polynomials pn x it is shown that the logarithmic derivative of the hankel determinant is characterized by a jimbo miwa okamoto form of the painleve vi system

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

on rubin s variant of the p adic birch and swinnerton dyer conjecture ii let e q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field k in 1991 by studying a certain special value of the katz two variable p adic l function lying outside the range of p adic interpolation k rubin formulated a p adic variant of the birch and swinnerton dyer conjecture when e k is infinite and he proved that his conjecture is true for e k of rank one when e k is finite however the statement of rubin s original conjecture no longer applies and the relevant special value of the appropriate p adic l function is equal to zero in this paper we extend our earlier work and give an unconditional proof of an analogue of rubin s conjecture when e k is finite

miura transformations for discrete painleve equations coming from the affine e8 weyl group we derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine weyl group e8 1 by deautonomisation we obtain two hitherto unknown systems one of which turns out to be a linearisable one and we show that both these systems arise from the deautonomisation of a non qrt quispel roberts thompson mapping in order to unambiguously prove the integrability of these nonautonomous systems we introduce a series of miura transformations which allows us to prove that one of these systems is indeed a discrete painleve equation related to the affine weyl group e7 1 and to cast it in canonical form a similar sequence of miura transformations allows us to effectively linearise the second system we obtain an interesting off shoot of our calculations is that the series of miura transformations when applied at the autonomous limit allows one to transform a non qrt invariant into a qrt one

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

fuchs solution of painleve vi equation by means of jacobi s last multiplier we use the known relationship between the lagrangian and jacobi s vorlesungen uber dynamik nebst funf hinterlassenen abhandlungen desselben herausgegeben von a clebsch druck und verlag von georg reimer berlin 1886 last multiplier of second order ordinary differential equations to determine two first integrals and their noether nachr ges wiss goettingen math phys kl 2 235 1918 symmetries for the fuch s solution c r acad sci 141 555 1905 of painleve vi after finding two lagrangians in addition to known hamiltonian given by malmquist ark mat astr fys 17 1 1922 the general solution follows and we show it to be the same as given by fuchs

the pointwise estimates of solutions for euler equations with damping in multi dimensions we study the time asymptotic behavior of solutions for the isentropic euler equations with damping in multi dimensions the global existence and pointwise estimates of the solutions are obtained furthermore we obtain the optimal lp 1 p convergence rate of the solution when it is a perturbation of a constant state our approach is based on a detailed analysis of the green function of the linearized system and some energy estimates

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

r matrix construction of electromagnetic models for the painleve transcendents the painleve transcendents pi pv and their representations as isomonodromic deformation equations are derived as nonautonomous hamiltonian systems from the classical r matrix poisson bracket structure on the dual space slr 2 of the loop algebra slr 2 the hamiltonians are obtained by composing elements of the poisson commuting ring of spectral invariant functions on slr 2 with a time dependent family of poisson maps whose images are four dimensional rational coadjoint orbits in slr 2 each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time varying electromagnetic field the painleve equations follow from reduction of these systems by the hamiltonian flow generated by a second commuting element in the ring of spectral invariants 1995 american institute of physics

the associated camassa holm equation and the kdv equation recently the associated camassa holm ach equation related to the fuchssteiner fokas camassa holm equation by a reciprocal transformation was introduced by schiff who derived backlund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions we show how the ach equation is related to schrodinger operators and kdv and describe how to construct solutions of ach from tau functions of the kdv hierarchy rational n soliton and elliptic solutions are considered as well as exact solutions given by a particular case of the third painleve transcendent

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

on tau functions for orthogonal polynomials and matrix models let v be a real polynomial of even degree and let rho be the equilibrium probability measure for v with support s to that v x greeater than or equal to int 2log x y rho dy c for some constant c with equality on s then s is the union of finitely many boundd intervals with endpoints delta j and rho is given by an algebraic weight w x on s then the system of orthogonal polynomials for w gives rise to a system of differential equations known as the schlesinger equations this paper identifies the tau function of this system with the hankel determinant det int x j k rho dx the solutions of the magnus schlesinger equation are realised by a linear system which is used to compute the tau functions in terms of a gelfand levitan equation the tau function is associated with a potential q and a scattering problem for the schrodinger equation with potential q the paper describes cases where this is integrable in terms of the nonlinear fourier transform

circular pentagons and real solutions of painleve vi equations we study real solutions of a class of painleve vi equations to each such solution we associate a geometric object a one parametric family of circular pentagons we describe an algorithm that permits to compute the numbers of zeros poles 1 points and fixed points of the solution on the interval 1 infty and their mutual position the monodromy of the associated linear equation and parameters of the painleve vi equation are easily recovered from the family of pentagons

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

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

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

the q painleve v equation and its geometrical description we study the q painleve v equation which can be obtained from the degeneration of the q pvi in the form of the asymmetric q piii equation and present its geometrical description based on the bilinear formulation we obtain the equations for the multi dimensional functions of q pv in the form of nonautonomous hirota miwa systems which lives in the weight lattice of the a4 affine weyl group this geometrical approach furnishes in a straightforward way the miuras and the schlesingers of q pv

euler systems and jochnowitz congruences this article relates the gross zagier formula with a simpler formula of gross for special values of l series via the theory of congruences between modular forms given two modular forms f and g of different levels which are congruent but whose functional equations have sign 1 and 1 respectively and an imaginary quadratic field k satisfying certain auxiliary conditions the main result gives a congruence between the algebraic part of l f k 1 expressed in terms of heegner points and the algebraic part of the special value l g k 1 congruences of this type were anticipated by jochnowitz and for this reason are referred to as jochnowitz congruences introduction let e be a modular elliptic curve over q of conductor n and let k be a quadratic imaginary field the l function l e k s has a functional equation of the standard kind relating its values at s and 2 s when all the primes dividing n are split in k then k is said to satisfy the heegner hypothesis relative to e in that case the sign in the functional equation of l e k s is 1 furthermore a construction of birch and heegner based on the theory of com plex multiplication gives a heegner point yk e e k coming from the modular parametrization

entropy and global existence for hyperbolic balance laws this paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables we propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state in addition we show that a system of balance laws satisfies a kawashima condition if and only if its first order approximation namely the hyperbolic parabolic system derived through the chapman enskog expansion satisfies the corresponding kawashima condition the result is then applied to bouchut s discrete velocity bgk models approximating hyperbolic systems of conservation laws

the fisher hartwig conjecture and generalizations abstract we discuss the status of the fisher hartwig conjecture concerning the asymptotic expansion of a class of toeplitz determinants with singular generating functions a counterexample is given for a nonrational generating function and we formulate a generalized fisher hartwig conjecture

painleve equations from darboux chains i piii pv we show that the painleve equations piii pvi can be derived in a unified way from a periodic sequence of darboux transformations for a schrodinger problem with quadratic eigenvalue dependence the general problem naturally divides into three different branches each described by an infinite chain of equations the painleve equations are obtained by closing the chain periodically at the lowest nontrivial level s the chains provide symmetric forms for the painleve equations from which hirota bilinear forms and lax pairs are derived in this paper part i we analyse in detail the cases piii pv while pvi will be studied in part ii