Finite-order solutions and the integrability of difference equations

The existence of sufficiently many finite-order (in the sense of Nevanlinna) mero-morphic solutions of a difference equation appears to be a good indicator of integra-bility. This criterion is used to single out dP II from a natural class of second-order difference equations. The proof given uses an estimate related to singularity confinement. The Painlevé property has been used as a detector of integrability for differential equations since the nineteenth century. An ordinary differential equation is said to possess the Painlevé property if all solutions are single-valued about all movable singularities (see, e.g., [1]). In Ablowitz, Halburd, and Herbst [2] the idea of using the complex analytic structure of solutions as a detector of integrability was extended from differential to difference equations. There it was suggested that many difference equations that are considered to be of " Painlevé type " admit sufficiently many finite-order (in the sense of Nevanlinna theory) meromorphic solutions. We will show that this property singles out dP II — Eq. (11) below, from a wide class of second-order difference equations. Eq. (11) has many properties associated with integrability and is known to possess a simple continuum limit to the second Painlevé (differential) equation, P II : y = 2y 3 + zy + σ, where σ is a constant. The singularity confinement approach to integrability of Grammaticos, Ramani, and Papageorgiou [3] has proved to be an easy to implement and quite powerful detector of integrability. It has led to the discovery of many important discrete equations [4] which are widely believed to be integrable. The approach involves studying the behaviour of iterates of finite initial conditions which lead to some future iterate becoming infinite. On continuing through the singularity, one finds that generically future iterates oscillate between finite and infinite values. Roughly speaking, a singularity is " confined " if the iterates return to finite values and contain enough information about the initial conditions. See also [5]. The idea of singularity confinement also presents a number of problems. In particular, how do we decide whether a given singularity sequence is truly confined and what exactly is the property for which we are testing? Also, an example of a numerically chaotic discrete equation possessing the singularity confinement property was found by Hietarinta and Viallet [6]. They suggest that singularity confinement needs to be augmented by a condition that a sequence of iterates possesses zero algebraic entropy. This is …