Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree

In this paper we construct all Painlevé-type differential equations of the form (d²y/dx²)² = F(x,y,dy/dx), where F is rational in y and y′=dy/dx, locally analytic in x, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear in y″ in the class of equations under investigation forces F to be a polynomial in y and y′. We find exactly six distinct classes of second-degree Painlevé equations, denoted SD-I,??,SD-VI, some of which further subdivide into canonical subcases. Only the first three classes (or at least equations transformable to the first three classes) and part of the sixth have appeared previously in the literature, especially the work of Chazy and Bureau. The fourth and fifth classes are new. The unified treatment of SD-I, which we call the “master Painlevé equation,” is new. Complete solutions are given in terms of the classical Painlevé transcendents, elliptic functions, or solutions of linear equations. In an appendix, it is shown that a class of second-degree equations generalizing the Appell equation can always be reduced to a second-order linear equation.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

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.     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

On Airy Solutions of the Second Painlevé Equation

In this paper, we discuss Airy solutions of the second Painlevé equation (P_II) and two related equations, the Painlevé XXXIV equation () and the Jimbo–Miwa–Okamoto σ form of P_II (S_II), are discussed. It is shown that solutions that depend only on the Airy function have a completely different structure to those that involve a linear combination of the Airy functions and . For all three equations, the special solutions that depend only on are tronquée solutions, i.e., they have no poles in a sector of the complex plane. Further, for both and S_II, it is shown that among these tronquée solutions there is a family of solutions that have no poles on the real axis.     

Painlevé VI and the Unitary Jacobi Ensembles

The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well‐known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin‐spin correlation functions of the two‐dimensional Ising model in the work of McCoy, Tracy, and Wu. The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double‐Wishart distribution. We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ‐form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .     

Parabolic First and Second Order Differential Equations

We are concerned with an inverse problem for a first-order linear evolution equation. Moreover, a complete second-order evolution equation will be considered, too. We indicate sufficient conditions for existence and uniqueness of a solution. All the results apply well to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

The Painlevé Property and Hirota's Method

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self-truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability

The characteristic feature of the so-called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly-Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.     

二阶中立型方程的强迫振动

讨论了一类具有偏差变元的二阶中立型方程,给出了该类方程解振动的充分条件.     

On One-Parameter Families of Painlevé III

Albrecht, Mansfield, and Milne developed a direct method with which one can calculate special integrals of polynomial type (also known as one-parameter family conditions, Darboux polynomials, eigenpolynomials, or algebraic invariant curves) for nonlinear ordinary differential equations of polynomial type. We apply this method to the third Painlevé equation and prove that for the generic case, the set of known one-parameter family conditions is complete.     

On Unstable Neutral Differential Equations of the Second Order

The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation (x(t)-px(t-))-q(t)x((t))=0 to be oscillatory and to improve some existing results. The main results are based on the comparison principles.     

The numerical solution of the second Painlevé equation

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second‐order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

二阶线性微分方程组解法研究

采用降阶法和欧拉方法对一类二阶线性微分方程组的求解进行了研究,并给出了当系数矩阵的特征根为三种不同情况（互异、共轭、二重根）时微分方程组的通解公式,并通过算例验证了通解的正确性.     

非线性二阶中立型微分方程的振动性

考虑中立型微分方程我们建立了方程(1)的若干振动准则。这些结果适用于强超线性和强次线性中立型微分方程,且当p(t)≡0时仍然成立。     

Nonoscillation Characterizations of Second Order Linear Differential Equations

In this paper, we establish some nonoscillatory characterizations of solutions of the second order linear differential equation (pu′)′ + qu = 0. Using these characterizations, we give some comparison theorems improving the results of Hille , Wintner , Wong and Yu .     

一类二阶微分方程解的振动性质

利用积分平均技巧研究二阶微分方程(r(t)(x(t) )x′(t) )′ q(t) f(x(t) ) g(x′(t) ) =0 .解的振动性质 ,得到了一些保证此方程所有解振动的充分条件 .特别 ,本文的结果改进了文 [1 ]的主要结果 .