We consider a family of solutions to the Painlevé II equation $$ u''(x)=2u^3(x)+xu(x)-α\qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously derive their singular asymptotics as $x \to -\infty$. In the meantime, we extend the existing asymptotic results when $x\to +\infty$ from $\a-\frac{1}{2} \notin \mathbb{Z}$ to any real $\a$. The connection formulas are also obtained.
In this paper, we establish a formula determining the value of the Cauchy integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for Ablowitz-Segur solutions of homogeneous Painlevé II equation.
We consider the second Painlevé equation $$ u"(x)=2u^3(x)+xu(x)-α, $$ where $α$ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as $x \to \pm \infty$ for both the real and purely imaginary Ablowitz-Segur solutions, as well as the corresponding connection formulas. We also show that the real Ablowitz-Segur solutions have no real poles when $α\in (-1/2, 1/2)$.
We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation $$ u"(x)=2u^3(x)+xu(x)-α\qquad \textrm{for } α\in \mathbb{R} \textrm{ and } |α| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|α| > 1/2$, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess $[ \, |α| + \frac{1}{2} \, ]$ simple poles on the real axis, which rigorously justifies the num…
A system of nonlinear differential equations $x^{1+γ}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by using the theory of Borel summable functions in asymptotic analysis, and apply results to Painlevé equations.
In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients…
In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier pa…
We show that there exists a rational change of coordinates of Painlevé's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singula…
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.
We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.
We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler--Bernoulli Beam equation which is the Painlevé--Ince Equation with additional terms. The equation possesses the same leading-order behaviour and resonances as the Painlevé--Ince Equation and has a Right Painlevé Series. However, it has no Left Painlev% é Series. A conjecture for the existence of Left Painlevé Series for ordinary differential equations is given.
Recently in a paper by Lin, Dai and Tibboel, it was shown that the third and fourth Painlevé equations have tronquée and tritronquée solutions. We obtain global information about these tronquée and tritronquée solutions. We find their sectors of analyticity, their Borel summed representations in these sectors as well as the asymptotic position of the singularities near the boundaries of the analyticity sectors. We also correct slight errors in the paper mentioned.
In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the Deift-Zhou steepest descent method. The asymptotic formulas of the leading coefficients and the recurrence coefficients for large n are described in terms of a class of analytic solutions to the the σ-form of the Painlevé II equation and the Painlevé XXXIV equation.
A unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized Borel summability. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the Painlevé property is present the equations are explicitly solvable and in the contrary case, under further assumptions, the integrals of mo…
The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces ${\mathbb C}P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux's coordinates are systematically studied in a unified way with the aid of the orbifold structure of ${\mathbb C}P^3(p,q,r,s)$ and dynamical systems theory.
The article studies the Fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity from the point of view of confluence from the Sixth Painlevé equation. This approach is developped separately on both sides of the Riemann-Hilbert correspondance. On the side of the nonlinear Painlevé-Okamoto foliation the relation between the nonlinear monodromy group of Painlevé VI and the "nonlinear wild monodromy pseudogroup" of Painlevé V (that is the pseudogroup generated by nonlinear monodromy, nonlinear Stokes operators and nonlinear exponential torus) is expl…
The Painlevé classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlevé analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of…
We consider a linear $2\times2$ matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as $P_6\to P_5$, where $P_n$ is the $n$-th Painlevé equatio…
In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quas…
We continue with the study of the Hankel determinant, $$ D_{n}(t,α):=\det\left(\int_{0}^{\infty}x^{j+k}w(x;t,α)dx\right)_{j,k=0 }^{n-1}, $$ generated by singularly perturbed Laguerre weight, $$ w(x;t,α):=x^α{\rm e}^{-x}\:{\rm e}^{-t/x}, \quad 0\leq x<\infty,\;\;\;α>0,\;\;\;\;t>0, $$ obtained through a deformation of the Laguerre weight function, $$ w(x;0,α):=x^α{\rm e}^{-x},\quad 0\leq x<\infty,\;\; α>0, $$ via the multiplicative factor ${\rm e}^{-t/x}$. \\ An earlier investigation was made on the finite $n$ aspect of the problem, this has appeared in \cite{ci1}. There, it was found that the l…
We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierar…
Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlevé equations are investigated. A tuple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. Each polynomial Painlevé equation has a regular weight. Conversely, for $2$ and $4$-dim cases, it is shown that there exists a differential equation satisfying the Painlevé property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical s…
We consider the generalized Painlevé--Ince equation, \begin{equation*} \ddot{x}+αx\dot{x}+βx^{3}=0 \end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $β=α^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlevé test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlevé--Ince equation fails at the Painlevé te…
The solutions of the perturbed first Painlevé equation $y"=6y^2-x^μ$, $μ>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the …
We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is applied to Painlevé's first equation. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions, obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain rest…
Higher dimensional analogs of the Painlevé equations have been proposed from various aspects. In recent studies, 4-dimensional analogs of the Painlevé equations were classified into 40 types. The aim of the present paper is to geometrically characterize these 40 types of equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. We obtain two functionally independent conserved quantities $H_1$ and $H_2$ for each system. We construct fibrations whose fiber at a general point $h_i $ is the spectral curve of the system with $H_i=h_i$ f…
We study the asymptotic behaviour of the solutions of the fifth Painlevé equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.
We review some aspects of recent work concerning double scaling limits of singularly perturbed hermitian random matrix models and their connection to Painlevé equations. We present new results showing how a Painlevé III hierarchy recently proposed by the author can be connected to the Lenard recursion formula used to construct the Painlevé I and II hierarchies.
In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlevé VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlevé VI equations. Instead of the system form, we especially focus on the Darboux equation in a scalar form, which is the generalization of the classical Lamé equation. We introduce a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) %around each of the four regul…
Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system associated with Painleve IV, as discussed in a recent paper), we give a number of results towards a classification of the real solutions of Painleve IV (or, more precisely, symmetric Painleve IV), according to their asymptotic behavior and singularities. We show the existence of globally nonsingular real solutions of symmetric Painleve IV for arbitrary nonzero v…
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^α e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $α>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of $α$, …
Rational solutions of the inhomogeneous Painleve-II equation and of a related coupled Painleve-II system have recently arisen in studies of fluid vortices and of the sine-Gordon equation. For the sine-Gordon application in particular it is of interest to understand the large-degree asymptotic behavior of the rational Painleve-II functions. We explicitly compute the leading-order large-degree asymptotics of these two families of rational functions valid in the whole complex plane with the exception of a neighborhood of a certain piecewise-smooth closed curve. We obtain rigorous error bounds by …
This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behavior. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve…
A local behavior of solutions of the Schlesinger equation is studied. We obtain expansions for this solutions, which converge in some neighborhood of a singular point. As a corollary the similar result for the sixth Painlevé equation was obtained. In our analysis, we use the isomonodromic approach to solve this problem.
We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conform…
We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlevé equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the matrix-valued function satisfying the associated Riemann-Hilbert problem can be represented by means of a local parametrix around the origin, whose existence can be proved by a vanishing lemma. The aim of this paper is to construct the explicit form of this parametrix and apply it to obtain improved asymptotic relations for the real and purely imaginary Ablowit…
We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.
In families of Painlevé VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case, we have one family with zero dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers done by Herfurtner in 1992 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete isomodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.
The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painlevé property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of $\C P^3(p,q,r,s)$. In particular, for the first Painlevé equation, a well known Painlevé's transformation is geomet…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlevé VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of Bäcklund transformations. For particular lines in the lattice, this proves conjectures of Bazhanov and Mangazeev. As applications, we describe the behaviour of the corresponding solutions near the singular points of Painlevé VI, and obtain several new properties of our polynomials.
In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlevé equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit f…
Modified Painlevé systems with meromorphic solutions for polynomial Hamiltonians of all degrees
Faculty of Mathematics Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学与机械学院）；School of Mathematics and Physics, University of Portsmouth（朴茨茅斯大学数学与物理学院）
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…
In this paper, we show that the family of moduli spaces of $\balpha'$-stable $(\bt, \blambda)$-parabolic $ϕ$-connections of rank 2 over $\BP^1$ with 4-regular singular points and the fixed determinant bundle of degree -1 is isomorphic to the family of Okamoto--Painlevé pairs introduced by Okamoto \cite{O1} and \cite{STT02}. We also discuss about the generalization of our theory to the case where the rank of the connections and genus of the base curve are arbitrary. Defining isomonodromic flows on the family of moduli space of stable parabolic connections via the Riemann-Hilbert correspondences…
We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor blades (or persistent singularities).
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t -> +/-\infty$ and $t -> +/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotical…
The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last …
Applications of the Painlevé-Kuratowski convergence: Lipschitz functions with converging Clarke subdifferentials and convergence of sets defined by converging equations
We study the hermitian one matrix model with semi-classical potential. This is a general unitary invariant random matrix ensemble in which the potential has a derivative that is a rational function and the measure is supported on some collection of disjoint closed intervals. Such models have attracted much interest both due to their physical applications and relations to integrable systems. An object of central interest in random matrix theory is the correlation kernel, as this encodes the eigenvalue correlation functions. In recent years many results have been obtained proving that the correl…
In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonloga…
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be $τ$-functions for Painlevé systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matri…
In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis.
