Rod Halburd

On the extension of the Painlevé property to difference equations

It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z ), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

The Painlev´ e property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlev´ e equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlev´ e property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z +1 )w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

Integrable systems and reductions of the self-dual Yang–Mills equations

Many integrable equations are known to be reductions of the self-dual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the classical Painleve equations and integrable generalizations of the Darboux–Halphen system and Chazy equations. The Chazy equation, first derived in 1909, is shown to correspond to the equations studied independently by Ramanujan in 1916.

The Generalized Chazy Equation from the Self‐Duality Equations

It is shown that classically known generalizations of the Chazy equation and Darboux–Halphen system are reductions of the self-dual Yang–Mills (SDYM) equations with an infinite-dimensional gauge algebra. The general ninth-order Darboux–Halphen system is reduced to a Schwarzian equation which governs conformal mappings of regions with piecewise circular sides. The generalized Chazy equation is shown to correspond to special mappings where either the triangles are equiangular or two of the angles are π/3.

How to detect the integrability of discrete systems

Several integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be described.

On the meromorphic solutions of an equation of Hayman

Abstract The behavior of meromorphic solutions of differential equations has been the subject of much study. Research has concentrated on the value distribution of meromorphic solutions and their rates of growth. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi-parameter ordinary differential equation introduced by Hayman. This equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions—only those that are meromorphic. This is achieved by combining Wiman–Valiron theory and local series analysis. Hayman conjectured that all entire solutions of this equation are of finite order. All meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one.

On Painlevé and Darboux-Halphen-Type Equations

It is now well known that a deep connection exists between soliton equations and ODEs of Painleve type. As a consequence there has been a significant reemergence of interest in the study of such ODEs and related issues. In this paper we demonstrate that a novel class of nonlinear ODEs, Darboux-Halphen (DH) type systems, can be obtained as reductions of the self-dual Yang-Mills (SDYM) equations. We show how to find by reduction from SDYM the associated linear pair for DH. This linear system is found to be monodromy evolving, which is different from the linear systems associated with the Painleve equations, which are isomonodromy. The solution of the DH system can be obtained in terms of Schwarzian equations, which are themselves linearizable. The DH system has solutions that are related to Painleve equations, but the solutions can have complicated analytic singularities such as natural boundaries and dense branching.

First integrals of a generalized Darboux–Halphen system

A third-order system of nonlinear, ordinary differential equations depending on three arbitrary parameters is analyzed. The system arises in the study of SU(2)-invariant hypercomplex manifolds and is a dimensional reduction of the self-dual Yang–Mills equation. The general solution, first integrals, and the Nambu–Poisson structure of the system are explicitly derived. It is shown that the first integrals are multi-valued on the phase space even though the general solution of the system is single-valued for special choices of parameters.

Movable algebraic singularities of second-order ordinary differential equations

Any nonlinear equation of the form y″=∑n=0Nan(z)yn has a solution with leading behavior proportional to (z−z0)−2/(N−1) about a point z0, where the coefficients an are analytic at z0 and aN(z0)≠0. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of z−z0. For these equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This generalizes results of Shimomura [“On second order nonlinear differential equations with the quasi-Painleve property II,” RIMS Kokyuroku 1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith [“On the singularities in the complex plane of the solutions of y″+y′f(y)+g(y)=P(x),” Proc. Lond. Math. Soc. 3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation out...

Solvable models of relativistic charged spherically symmetric fluids

It is known that charged relativistic shear-free fluid spheres are described by the equation yxx = f(x)y2 + g(x)y3, where f and g are arbitrary functions of x only and y is a function of x and an external parameter t. Necessary and sufficient conditions on f and g are obtained such that this equation possesses the Painleve property. In this case the general solution y is given in terms of solutions of the first or second Painleve equation (or their autonomous versions) and solutions of their linearizations. In the autonomous case we recover the solutions of Wyman, Chatterjee and Sussman and a large class of (apparently new) solutions involving elliptic integrals of the second kind. Solutions arising from the special Airy function solutions of the second Painleve equation are also given. It is noted that, as in the neutral case, a three-parameter family of choices of f and g are described by solutions of an equation of Chazy type.