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.     

Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.     

Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be an admissible finite-order meromorphic solutionof the second-order difference equation

where R(z, w(z)) is rational in w(z) withcoefficients that are meromorphic in z. Then either w(z) satisfiesa difference linear or Riccati equation or else the above equationcan be transformed to one of a list of canonical differenceequations. This list consists of all known difference Painlevéequations of the above form, together with their autonomousversions. This suggests that the existence of finite-order meromorphicsolutions is a good detector of integrable difference equations.     

On the meromorphic solutions of an equation of Hayman

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.     

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie lemma, as well as other results on the value distribution of solutions.     

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.