Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painlevé-type differential equations of the second order and second degree that are built upon, in a natural well-defined sense, the "sn-log" equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling "exotic" Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé-type equations contains three, and only three, equations, which we denote SD-326-I, SD-326-II, and SD-326-III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn-log equation. The second and third are new, the third being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second-degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.     

Binomial-Type Ordinary Differential Equations of the Third Order

The complete Painlevé classification of the binomial ordinary differential equations of the third order is built. Eight classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Higher-Order Painlevé Equations in the Polynomial Class II: Bureau Symbol P1

In this article, we complete the Painlevé classification of fourth-order differential equations in the polynomial class that was begun in paper I, where the subcase having Bureau symbol P 2 was treated. This article treats the more difficult subcase having Bureau symbol P 1. Some of the calculations involve the use of computer searches to find all cases of integer resonances. Other cases are better handled with the Conte–Fordy–Pickering test for negative resonances. The final list consists of 19 equations denoted F-I, F-II, … , F-XIX, 17 of which have the Painlevé property while 2 (F-II, F-XIX) have Painlevé violations but are nevertheless very interesting from the point of view of Painlevé analysis. The main task of this article is to prove that the 17 Painlevé-type equations and the equivalence classes that they generate provide the complete classification of the fourth-order polynomial class. Equations F-V, F-VI, F-XVII, and F-XVIII define higher-order Painlevé transcendents. Of these, F-VI was new in paper I while the other three are group-invariant reductions of the KdV5, the modified KdV5, and the modified Sawada–Kotera equations, respectively. Seven of the 19 equations involve hyperelliptic functions of genus 2. Partial results on the fourth-order classification problem have been obtained previously by Bureau, Exton, and Martynov, the latter author finding all but four of the relevant reduced equations. Complete solutions are given except in the cases that define the aforementioned higher-order transcendents.     

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painlevé equation.     

The Mirror Systems of Integrable Equations

We provide an algorithm to convert integrable equations to regular systems near noncharacteristic, movable singularity manifolds of solutions. We illustrate how the algorithm is equivalent to the Painlevé test. We also use thealgorithm to prove the convergence of the Laurent series obtained from the Painlevé test.     

Higher-order Painlevé Equations in the Polynomial Class I. Bureau Symbol P2

In this article, we construct all fourth- and fifth-order differential equations in the polynomial class having the Painlevé property and having the Bureau symbol P 2. The fourth-order equations (including the Bureau barrier equation, y ^(iv)=3 yy "−4( y ')², which fails some Painlevé tests) are six in number and are denoted F-I,…,F-VI; the fifth-order equations are four in number and are denoted Fif-I,…,Fif-IV. The 12 remaining equations of the fourth order in the polynomial class (where the Bureau symbol is P 1) are listed in the Appendix, their proof of uniqueness being postponed to a sequel (paper II). Earlier work on this problem by Bureau, Exton, and Martynov is incomplete, Martynov having found 13 of the 17 distinct reduced equations. Equations F-VI and Fif-IV are new equations defining new higher-order Painlevé transcendents. Other higher-order transcendents appearing here may be obtained by group-invariant reduction of the KdV5, Sawada–Kotera, and Kaup–Kupershmidt equations, the latter two being related. Four sections are devoted to solutions, first integrals, and assorted properties of the main equations. Several of the equations are solved in terms of hyperelliptic functions of genus 2 by means of Jacobi's postmultiplier theory. Except for a classic solution of Drach, we believe that all of these hyperelliptic solutions are new. In an accompanying paper, the hyperelliptic solutions of F-V and F-VI are applied to the unsolved third-order Chazy classes IX and X.     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

Symmetries and Large Time Asymptotics of Compressible Euler Flows with Damping

Large time asymptotics of compressible Euler equations for a polytropic gas with and without the porous media equation are constructed in which the Barenblatt solution is embedded. Invariance analysis for these governing equations are carried out using the classical and the direct methods. A new second order nonlinear partial differential equation is derived and is shown to reduce to an Euler–Painlevé equation. A regular perturbation solution of a reduced ordinary differential equation is determined. And an exact closed form solution of a system of ordinary differential equations is derived using the invariance analysis.     

The General Theory of Linear Difference Equations over the Max-Plus Semi-Ring

We present the mathematical theory underlying systems of linear difference equations over the max-plus semi-ring. The result provides an analog of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q -difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric P_IV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.     

Chazy Classes IX–XI Of Third-Order Differential Equations

In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third-order differential equations in the polynomial class having the Painlevé property. Classes IX and X are the only Chazy classes that have remained unsolved to this day, and they have been at the top of our "most wanted" list for some time. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.) The method involves transforming to a more tractable class of fourth- and fifth-order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher-order Painlevé transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents. In an appendix, we present the full list of Chazy equations (in the third-order polynomial class) and the solutions of those that are not dealt with in the body of this article.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

In this article our concern is with the third Painlevé equation
d² y /d x ²= (1/ y )(d y /d x )²− (1/ x )(d y /d x ) + ( αy ²+ β )/ x + γy ³+ δ / y
where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x ^1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.     

Justification of Asymptotic Formulas for the Fourth Painlevé Equation

A special case of the fourth Painlevé equation is studied. The existence theorem is proved, and asymptotic formulas for the two parametric family of solutions near negative infinity are obtained.     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.     

Toward the Exact WKB Analysis of the PII Hierarchy

We develop the exact WKB analysis of the P _II hierarchy introduced by Gordoa et al. [ 5 ] in this paper; our particular interest is in the relation between the Stokes geometry of a higher order Painlevé equation in the hierarchy and that of its underlying Lax pair. An important observation is that Stokes curves of the Painlevé equation may cross, reflecting the higher order character of the equation, and in a neighborhood of the crossing point an unexpected degeneracy of the Stokes geometry of the Lax pair may occur along a curved ray emanating from the crossing point.     

第一类Painlevé方程的高阶类似(4P_1)的解析性质

该文研究了第一类Painlevé方程高阶类似(4P_1)的超越亚纯解的解析性质并获得一系列结果     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

Direct Similarity Analysis of Generalized Burgers Equations and Perturbation Solutions of Euler–Painlevé Transcendents

Similarity reductions of the generalized Burgers equation     , where α, β, and γ are non-negative constants, n a positive integer and   j = 0, 1, 2  , are obtained by the direct method of Clarkson and Kruskal [ 1 ]. This is the first work to report the similarity variables as an incomplete gamma function and also as a power of     , and to provide a perturbation solution of an Euler–Painlevé transcedent.