Christopher M. Cosgrove

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 Painleve 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 Chazys 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 Painleve 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.

New family of exact stationary axisymmetric gravitational fields generalising the Tomimatsu-Sato solutions

A new three-parameter family of exact solutions of the stationary axisymmetric vacuum Einstein equations, which represent rotating bounded sources, are presented. This family contains the solutions of Kerr and Tomimatsu-Sato as special cases, and may be regarded as a generalisation of the latter to arbitrary continuous delta parameter. The final form of the metric depends on two ordinary differential equations of the second order. When delta is not an integer, these equations define unfamiliar transcendental functions for which rapidly converging series expansions of several types are available. When delta is an integer, the solutions are polynomial or rational functions of spheroidal coordinates and define the discrete Tomimatsu-Sato series, for which those authors give the cases delta =1, 2, 3, 4.

Eigenvalues of the Chandrasekhar–Page angular functions

The Chandrasekhar–Page angular functions for the Dirac equation in the Kerr–Newman background are expanded as series of hypergeometric polynomials, and a three‐term recurrence relation is derived for the coefficients in these series. This leads to a transcendental equation for the determination of the separation constant which is obtained initially as a power series and is then iterated by the method of Blanch and Bouwkamp.

A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. I. General theory

A new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented. The basic variable is e2γ = -g11g44 and satisfies a field equation of the fourth differential order which may be interpreted as the condition that a certain 2-space has constant curvature, K = -1. The principal motivation is that for many known solutions and all known asymptotically flat (non-static) solutions, e2γ takes a much simpler functional form than either the metric coefficients, g44, g34 and g33, or the Ernst potentials, E and ξ . Three methods are given for the construction of the full metric from e2γ. A duality principle is invoked to provide a very similar field equation for the metric coefficient, e2γ-2u = -g11.

Gap probabilities for edge intervals in finite Gaussian and Jacobi unitary matrix ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite-dimensional N ? N random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second- and third-order nonlinear ordinary differential equations defining the probabilities in the general N case. For N = 1 and 2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as N????, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a Painlev?-V transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a Painlev?-VI transcendent.

Chazy's second-degree Painlevé equations

We examine two sets of second-degree Painlev? equations derived by Chazy in 1909, denoted by systems (II) and (III). The last member of each set is a second-degree version of the Painlev?-VI equation, and there are no other second-order second-degree Painlev? equations in the polynomial class with this property. We map the last member of system (II) into the Fokas?Yortsos equation and demonstrate how both Schlesinger and Okamoto transformations for Painlev?-VI can be read off the Chazy equation. The 24 fundamental Schlesinger transformations were known to Garnier in 1943 while the 64 Okamoto transformations date from 1987. In an appendix, we gather together the solutions of the five members of system (II). System (III) is better known, being equivalent to Jimbo and Miwas equations for the logarithmic derivatives of the tau functions of the six Painlev? transcendents. The last member, known to Painlev? in 1906, was written in a manifestly symmetric form by Jimbo and Miwa, suggesting many induced symmetries for Painlev?-VI. In particular, Schlesinger and Okamoto transformations for Painlev?-VI can be read off immediately.

A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. II. Separable solutions

For pt.I see ibid., vol.11, no.12 (1978). The techniques of the preceding paper are applied to several cases where the gamma equation may be solved by separation of variables in the form, gamma = gamma 1( rho )+ gamma 2( tau ), where gamma 1( rho ) is either zero or a very simple function and gamma 2( tau ) satisfies an ordinary differential equation of the fourth order. Among the exact solutions constructed are the full six-parameter family of generalised Tomimatsu-Sato solutions, the rotating Curzon solution, the Kinnersley-Kelley solution and a class of solutions recently found by Ernst. Two new classes of solutions are presented as well as several new particular solutions expressible in closed form. All stationary axisymmetric vacuum metrics with a non-trivial second-rank Killing tensor whose components do not depend on the ignorable co-ordinates, phi and t, are derived. This problem reduces to finding separable solutions of the dual of the gamma equation of the form, e2 gamma -2u=R( rho , tau )(f( rho )+g( tau )), in four special co-ordinate systems, ( rho , tau ), where R( rho , tau ) is a prescribed simple function. A comparison is made with the canonical Schrodinger separable metric forms of Carter.

Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painleve-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 Painleve, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painleve in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painleve analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painleve 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” Painleve equations, which are Painleve equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painleve-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 Painleves 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 Painleve classification problems, including full generalizations of two other second-degree equations discovered by Painleve, additional examples of exotic Painleve equations and Painleve equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.

Integrability, random matrices and Painlevé transcendents

The probability that an interval I is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval I includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painleve transcendents.

On the initial value problem for the modified Benjamin–Ono equation

The initial value problem for the modified Benjamin–Ono equation is solved by exploiting the Fokas–Ablowitz inverse scattering transform for a certain complex extension of the Benjamin–Ono equation. The asymptotics of the Jost functions for small (positive) values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case.