moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

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

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

a finite element approach for vector and tensor valued surface pdes abstract we derive a cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on riemannian manifolds this allows to reformulate any vector and tensor valued surface pde in a form suitable to be solved by established tools for scalar valued surface pdes we consider piecewise linear lagrange surface finite elements on triangulated surfaces and validate the approach by a vector and a tensor valued surface helmholtz problem on an ellipsoid we experimentally show optimal linear order of convergence for these problems the full functionality is demonstrated by solving a surface landau de gennes problem on the stanford bunny all tools required to apply this approach to other vector and tensor valued surface pdes are provided

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

gap probability of the circular unitary ensemble with a fisher hartwig singularity and the coupled painleve v system we consider the circular unitary ensemble with a fisher hartwig singularity of both jump type and root type at z 1 a rescaling of the ensemble at the fisher hartwig singularity leads to the confluent hypergeometric kernel by studying the asymptotics of the toeplitz determinants we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the hamiltonian of the coupled painleve v system in dimension four this leads to a painleve type representation of the confluent hypergeometric kernel determinant moreover the large gap asymptotics including the constant terms are derived by evaluating the total integral of the hamiltonian in particular we reproduce the large gap asymptotics of the confluent hypergeometric kernel determinant obtained by deift krasovsky and vasilevska and the sine kernel determinant as a special case including the constant term conjectured earlier by dyson

a critical phenomenon in the two matrix model in the quartic quadratic case we study a critical behavior for the eigenvalue statistics in the two matrix model in the quartic quadratic case for certain parameters the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support the main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point the kernel is expressed in terms of a 4 4 riemann hilbert problem related to the hastings mcleod solution of the painleve ii equation we then compare the new kernel with two other critical phenomena that appeared in the literature before first we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit second we briefly discuss the relation with the tacnode singularity in noncolliding brownian motions that was recently analyzed although the limiting density in that model also vanishes like a square root at a certain interior point the process at the local scale is different from the process that we obtain in the two matrix model

convergence to traveling waves with decay rates for solutions of the initial boundary problem to a relaxation model abstract in this paper we consider a 2 2 relaxation hyperbolic system of conservation laws with a boundary effect and we show that the solutions of this initial boundary problem tend to the traveling wave solutions of the corresponding cauchy problem time asymptotically in particular we give the algebraic and exponential decay rates by using the weighted energy method the location of a shift for the traveling wave to overcome the difficulty in the boundary plays a key role in this paper

meshfree methods on manifolds for hydrodynamic flows on curved surfaces a generalized moving least squares gmls approach abstract we utilize generalized moving least squares gmls to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds we use exterior calculus to formulate incompressible hydrodynamic equations in the stokesian regime and handle the divergence free constraints via a generalized vector potential this provides less coordinate centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth order governing equations in terms of a system of second order elliptic operators using a hodge decomposition we develop methods for manifolds having spherical topology we show the methods exhibit high order convergence rates for solving hydrodynamic flows on curved surfaces the methods also provide general high order approximations for the metric curvature and other geometric quantities of the manifold and associated exterior calculus operators the approaches also can be utilized to develop high order solvers for other scalar valued and vector valued problems on manifolds

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

approximation of tensor fields on surfaces of arbitrary topology based on local monge parametrizations abstract we introduce a new method the local monge parametrizations lmp method to approximate tensor fields on general surfaces given by a collection of local parametrizations e g as in finite element or nurbs surface representations our goal is to use this method to solve numerically tensor valued partial differential equations pdes on surfaces previous methods use scalar potentials to numerically describe vector fields on surfaces at the expense of requiring higher order derivatives of the approximated fields and limited to simply connected surfaces or represent tangential tensor fields as tensor fields in 3d subjected to constraints thus increasing the essential number of degrees of freedom in contrast the lmp method uses an optimal number of degrees of freedom to represent a tensor is general with regards to the topology of the surface and does not increase the order of the pdes governing the tensor fields the main idea is to construct maps between the element parametrizations and a local monge parametrization around each node we test the lmp method by approximating in a least squares sense different vector and tensor fields on simply connected and genus 1 surfaces furthermore we apply the lmp method to two physical models on surfaces involving a tension driven flow vector valued pde and nematic ordering tensor valued pde on different topologies the lmp method thus solves the long standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom

level spacing functions and the connection problem of a fifth painleve transcendent in the study of level spacing functions for the eigenvalues of random matrices mahoux and mehta 1993 studied the functions s t a t and b t related to certain fredholm determinants these functions can be expressed in terms of the fifth or the third painleve transcendents the asymptotic behaviour for t to infinity of these functions and their derivatives with respect to a parameter were derived by them except for an unknown constant using jimbo s 1982 method of monodromy preserving deformations we connect the behaviour of the fifth painleve transcendent at t 0 and at t infinity thus determining the unknown constant

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

nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials we study polynomials orthogonal on a uniform grid we show that each weight function gives two potentials and each potential leads to a structure relation lowering operator these results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three term recurrence relations

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

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

the lp stability of relaxation rarefaction profiles we consider the large time behavior of solutions for a hyperbolic relaxation system for a certain class of initial data the solution is shown to converge to relaxation rarefaction profiles at a determined asymptotic rate the result is established without the smallness conditions of the wave strength and the initial disturbances 2001 academic press

hermite pade approximants for a pair of cauchy transforms with overlapping symmetric supports hermite pade approximants of type ii are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order we are interested in the situation when the approximated vector is given by a pair of cauchy transforms of smooth complex measures supported on the real line the convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals angelesco systems or two intervals that coincide under the condition that the ratio of the measures is a restriction of the cauchy transform of a third measure nikishin systems in this work we consider the case where the supports form two overlapping intervals in a symmetric way and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap we derive szeg type formulae for the asymptotics of the approximants identify the convergence and divergence domains the divergence domains appear for angelesco systems but are not present for nikishin systems and show the presence of overinterpolation a feature peculiar for nikishin systems but not for angelesco systems our analysis is based on a riemann hilbert problem for multiple orthogonal polynomials the common denominator 2016 wiley periodicals inc

nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions abstract in this paper we show that a strong planar rarefaction wave is nonlinear stable namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions compared with former results obtained by t p liu 1987 comm math phys 108 153 175 and t luo 1997 j differential equations 133 255 279 our main novelty lies in the fact that the planar rarefaction waves do not need to be small and in the one dimensional case the initial disturbance can also be chosen arbitrarily large

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

decay rate for travelling waves of a relaxation model abstract a relaxation model was proposed in shi jin and zhouping xin comm pure appl math 48 1995 555 563 to approximate the hyperbolic systems numerically under the subcharacteristic condition introduced in t p liu comm math phys 108 1987 153 175 the stability of travelling waves with strong shock profile and integral zero was proved in h l liu j h wang and t yang stability in a relaxation model with nonconvex flux preprint 1996 h l liu and j wang asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms preprint 1995 when the original system is scalar in this paper we study the rate of the asymptotic convergence speed of thse travelling wave solutions the analysis applies to the case of a nonconvex flux and when the shock speed coincides with characteristic speed of the state at infinity the decay rate is obtained by applying the energy method and is shown to be the same as the one for the viscous conservation law a matsumura and k nishihara comm math phys 165 1994 83 96

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

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

critical edge behavior in the modified jacobi ensemble and the painleve v transcendents we study the jacobi unitary ensemble defined by the perturbed chebyshev weight w x 1 x2 12 tn2 x2 x 1 1 tn 1 the algebraic singularity coalesces with the end point of the support of the weight function namely tn 1 the hard edge as n we use the riemann hilbert approach the method of deift and zhou to obtain the asymptotics of the polynomials orthogonal with respect to w x we show that the universality property is preserved the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel the main result is on the local behavior at the edge of the spectrum the limit kernel at the edge when tn 1 o n 2 is described by the function for a specific solution of the painleve v equation we also show that when tn varies to a fixed t 1 the painleve v kernel transits to the bessel kernel j 12 on the other hand when tn approaches 1 the painleve v kernel can be approximated by another bessel kernel j 12

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

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

asymptotics of a τ function arising in the two dimensional ising model the short distance assymptotics of the function associated to the 2 point function of the two dimensional ising model is computed as a function of the integration constant defined from the long distance behavior of the function the result is expressible in terms of the barnes double gamma function equivalently the barnesg function

the jin xin relaxation approximation of scalar conservation laws in several dimensions with large initial perturbation abstract this paper is concerned with nonlinear stability of strong planar rarefaction waves for the jin xin relaxation approximation of scalar conservation laws in several dimensions for such a problem local stability of weak or strong planar rarefaction waves have been obtained in luo 1997 20 and zhao 2000 43 respectively for the global stability results to the best of our knowledge the only result available now is on the one dimensional case cf zhao 2000 43 which is based on the maximum principle established in natalini 1996 30 the main purpose of this paper is try to deduce some nonlinear stability results with large initial perturbation our analysis is based on the elementary energy method and the continuation argument

convergence rates to the discrete travelling wave for relaxation schemes this paper is concerned with the asymptotic convergence of nu merical solutions toward discrete travelling waves for a class of relaxation nu merical schemes approximating the scalar conservation law it is shown that if the initial perturbations possess some algebraic decay in space then the nu merical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time provided the sums of the initial perturbations for the u component equal zero a polynomially weighted l 2 norm on the perturba tion of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate

zero relaxation limit to centered rarefaction waves for jin xin relaxation system abstract in this paper we study the zero relaxation limit problem for the following jin xin relaxation system e u t v x 0 v t a 2 u x 1 f u v with initial data i u v x 0 u 0 x v 0 x u v u 0 as x this system was proposed by jin and xin 1995 1 with an interesting numerical origin as the relaxation time tends to zero this system converges to the equilibrium conservation law formally our interest is to study the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the equilibrium conservation law contain centered rarefaction waves and the limits u v of the initial data at x do not satisfy the equilibrium equation i e v f u in particular riemann data connected by rarefaction curves are included we show that if the wave strength is sufficiently small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

a riemann problem based method for solving compressible and incompressible flows a riemann problem based method for solving two medium flow including compressible and incompressible regions is presented the material interface is advanced by front tracking method and the material interface boundary conditions are defined by modified ghost fluid method a coupled compressible and incompressible riemann problem constructed in the normal direction of the material interface is proposed to predict the interfacial states with the ghost fluid states the compressible and incompressible flows are solved by discontinuous galerkin method an incompressible discontinuous galerkin method with nonuniform time step is also deduced for shock wave formed in compressible flow the numerical errors for the ghost fluid method in earlier works are analyzed and discussed in the numerical examples it shows that the proposed method can provide reasonable results including shock wave location two medium flow including compressible and incompressible regions is investigated an incompressible discontinuous galerkin method with nonuniform time step is derived a coupled compressible and incompressible riemann problem is proposed reasonable results are obtained for strong shock wave formed in compressible flow

asymptotics for toeplitz determinants perturbation of symbols with a gap we study the determinants of toeplitz matrices as the size of the matrices tends to infinity in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate we generalize an asymptotic expansion by widom indiana univ math j 21 277 283 1971 which was known for symbols supported on an arc we highlight applications of our results in the circular unitary ensemble and in the study of fredholm determinants associated to the sine kernel

painleve ii asymptotics near the leading edge of the oscillatory zone for the korteweg de vries equation in the small dispersion limit in the small dispersion limit solutions to the korteweg de vries equation develop an interval of fast oscillations after a certain time we obtain a universal asymptotic expansion for the korteweg de vries solution near the leading edge of the oscillatory zone up to second order corrections this expansion involves the hastings mcleod solution of the painleve ii equation we prove our results using the riemann hilbert approach c 2009 wiley periodicals inc

toeplitz matrices and determinants with fisher hartwig symbols abstract this paper is concerned with toeplitz matrices generated by symbols of the form a t r 1 r t t r 2 r t t r r b t t 1 where t 1 t r are pairwise distinct points on the unit circle b is sufficiently smooth b t 0 t 1 and ind b 0 t t r r is defined as exp i r arg t t r with arg t t r and r r are complex numbers satisfying re r 1 2 re r 1 2 weighted hardy spaces l 2 1 and l 2 2 are defined such that the toeplitz operator t a l 2 1 l 2 2 is bounded and invertible and that the finite section method is applicable to t a considered as acting from l 2 1 onto l 2 2 these results are then applied to prove a conjecture of fisher and hartwig adv in chem phys 15 1968 333 353 which asserts that the determinants of the n n sections of t a are asymptotically equal to g b n n q e a n where q r 2 r 2 and g b e a are certain nonzero constants under quite general conditions on the smoothness of b these constants are completely identified

toeplitz determinants with merging singularities we study asymptotic behavior for determinants of n n toeplitz matrices corresponding to symbols with two fisher hartwig singularities at the distance 2t 0 from each other on the unit circle we obtain large n asymptotics which are uniform for 0 t t0 where t0 is fixed they describe the transition as t 0 between the asymptotic regimes of 2 singularities and 1 singularity the asymptotics involve a particular solution to the painlev e v equation we obtain small and large argument expansions of this solution as applications of our results we prove a conjecture of dyson on the largest occupation number in the ground state of a one dimensional bose gas and a conjecture of fyodorov and keating on the second moment of powers of the characteristic polynomials of random matrices

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

emergence of a singularity for toeplitz determinants and painleve v we obtain asymptotic expansions for toeplitz determinants corresponding to a family of symbols depending on a parameter t for t positive the symbols are regular so that the determinants obey szeg s strong limit theorem if t 0 the symbol possesses a fisher hartwig singularity letting t 0 we analyze the emergence of a fisher hartwig singularity and a transition between the two different types of asymptotic behavior for toeplitz determinants this transition is described by a special painleve v transcendent a particular case of our result complements the classical description of wu mccoy tracy and barouch of the behavior of a 2 spin correlation function for a large distance between spins in the two dimensional ising model as the phase transition occurs

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

lp convergence rate to diffusion waves for p system with relaxation in this paper we consider the asymptotic behavior of solution for the cauchy problem for p system with relaxation vt ux 0 ut p v x 1 f v u e with initial data v u x 0 v0 x u0 x v0 u as x i by applying the elementary energy method and the green function method for the parabolic equation we obtain the l p convergence rate to the diffusion waves for the cauchy

asymptotics of toeplitz determinants and the emptiness formation probability for the xy spin chain we study an asymptotic behaviour of a special correlator known as the emptiness formation probability efp for the one dimensional anisotropic xy spin 1 2 chain in a transverse magnetic field this correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models for the xy spin chain the correlator can be expressed as the determinant of a toeplitz matrix and its asymptotical behaviours for n throughout the phase diagram are obtained using known theorems and conjectures on toeplitz determinants we find that the decay is exponential everywhere in the phase diagram of the xy model except on the critical lines i e where the spectrum is gapless in these cases a power law prefactor with a universal exponent arises in addition to an exponential or gaussian decay the latter gaussian behaviour holds on the critical line corresponding to the isotropic xy model while at the critical value of the magnetic field the efp decays exponentially at small anisotropy one has a crossover from the gaussian to the exponential behaviour we study this crossover using the bosonization approach