Existence of finite-order meromorphic solutions as a detector of integrability in difference equations

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Abstract

The existence of sufficiently many finite-order (in the sense of Nevanlinna) meromorphic solutions of a difference equation appears to be a good indicator of integrability. It is shown that, out of a large class of second-order difference equations, the only equation that can admit a sufficiently general finite-order meromorphic solution is the difference Painlevé II equation. The proof given relies on estimates obtained by arguments related to singularity confinement. The existence of meromorphic solutions of a general class of first-order difference equations is also proven by a simple method based on Banach’s fixed point theorem.

Introduction

An ordinary differential equation is said to possess the Painlevé property if all of its solutions are single-valued about all movable singularities (see, e.g., [1]). This property is a powerful indicator of integrability of a differential equation. In the early twentieth century, Painlevé [2], [3], Fuchs [4] and Gambier [5] showed that, out of a general class of second-order ordinary differential equations, there were six equations possessing the Painlevé property which could not be integrated in terms of known functions. These equations are now known as the six Painlevé equations. The proof that these equations are indeed integrable had to wait until the latter part of the twentieth century when they were solved by inverse scattering techniques based on an associated isomonodromy problem; see, e.g., [6].

Several analogues of the Painlevé property for discrete equations have been discussed in the literature. In particular, Ablowitz et al. [7] considered discrete equations as delay equations in the complex plane which enabled them to utilize complex analytic methods. The equations they consider to be of “Painlevé type” (i) are of finite order in Nevanlinna theoretic sense, and (ii) have no digamma functions in their series expansions. They looked at, for instance, difference equations of the type y¯+y¯=R(z;y), where R is rational in both of its arguments and we have suppressed the z-dependence by writing yy(z), y¯y(z+1) and y¯y(z1). Ablowitz, Halburd and Herbst showed that if (1.1) has at least one non-rational finite-order meromorphic solution, then degy(R)2. Indeed, the difference Painlevé II (dPII) equation (3.21) lies within this class of equations. The assumption is considerably weaker than the continuous Painlevé property where all solutions are required to be single-valued about movable singularities. On the other hand, the class (1.1) with degy(R)2 also includes many equations generally considered to be non-integrable.

We will show that the existence of a sufficiently general finite-order meromorphic solution is enough to single out the difference Painlevé II equation from a general class (1.1) of difference equations; see Theorem 3.1 below. We will also give examples of non-Painlevé type difference equations having special finite-order Riccati solutions. Therefore demanding the existence of a finite number (or even a one-parameter family) of finite-order solutions is not always enough to single out the dPII from (1.1).

Costin and Kruskal [8] also applied complex analytic methods to detect integrability in discrete equations. Their approach is based on embedding solutions of a discrete equation into analyzable functions and is quite different from the property described in the present paper.

The most widely used detector of integrable discrete analogues of the Painlevé equations is the singularity confinement test of Grammaticos et al. [9]. The basic idea is to consider finite initial conditions leading to iterates which become infinite at a certain point. Such a singularity is said to be confined if the iterates are all finite after a finite number of steps and contain sufficient information about the initial conditions. The singularity confinement test has been successfully applied to discover many important discrete equations, which are widely believed to be integrable [10].

We will illustrate singularity confinement using the standard example [10]yn+1+yn1=anyn+bn1yn2, where (an) and (bn) are given sequences. To go from finite values of yn for nk to yk+1=, we must have yk=±1. In order to analyze future iterates we let yk1 have an arbitrary finite value c and we set yk=±1+ϵ and then take the limit ϵ0. The next few iterates are yk+1=12(akbk)ϵ1+O(1),yk+2=1+2ak+1akbkak±bkϵ+O(ϵ2),yk+3=±(ak±bk){(bk+2bk)(ak+22ak+1+ak)}2(2ak+1akbk)ϵ1+O(1). In order for yk+3 to be finite (i.e., if the singularity is confined) we must have (bk+2bk)(ak+22ak+1+ak)=0. If all singularities are confined then this condition must be true with both the “ +” and “ −” signs and for all k, in which case ak=αn+β and bn=γ+δ(1)n and Eq. (1.2) is the discrete Painlevé II equation.

Despite the demonstrated power and apparent simplicity of the singularity confinement test, its implementation is not without difficulty. In general the iterates of a difference equation oscillate between finite and infinite values. Therefore it is not always clear when the iterates truly leave the singularity. To be absolutely sure that the singularity is confined we would need to know information about infinitely many iterates. Another difficulty is that there exists a discrete equation discovered by Hietarinta and Viallet [11], which passes the singularity confinement test, but numerical studies suggest that it is chaotic. Their suggestion to avoid this problem is to demand that the iteration sequence has zero algebraic entropy. This approach is related to a number of techniques which use the slow growth of the degree of the nth iterate of the considered map (as a rational function of the initial conditions) to detect integrability [12], [13], [14], [15].

In this paper we consider Eq. (1.1) using the notion of singularity confinement involving only five iterates. Nevanlinna theory is used to demonstrate that generic non-confinement of poles implies that the order of any corresponding meromorphic solution is infinite. This helps to clarify a link between the complex analytic approach [7] and the singularity confinement test [9]. The latter is used as a tool in the proof of Theorem 3.1 to show that the only difference equation of the type (1.1) having a sufficiently general finite-order solution is the difference Painlevé II equation.

The autonomous form of the difference Painlevé II equation, known as the McMillan map (3.25), admits a two-parameter family of finite-order meromorphic solutions. In the non-autonomous case, however, the question of existence of meromorphic solutions remains open.

In general, very little is known about the singularity structure of solutions of difference equations in the complex domain. We conclude this paper by bringing together a number of existence results for meromorphic solutions of first-order autonomous difference equations. We prove these results using Banach’s fixed point theorem.

Section snippets

Nevanlinna theory

The Nevanlinna theory of meromorphic functions studies the density of the points in the complex plane at which a meromorphic function takes a prescribed value. It also provides a natural way to describe the growth of a meromorphic function. In this section we will first present some of the basic definitions and elementary facts from Nevanlinna theory. Then we will go on to prove a technical lemma, which will be applied in Section 3 to single out the difference Painlevé II equation from a

Singularity confinement and value distribution

The aim of this section is to prove Theorem 3.1, which uses value distribution theory to single out the difference Painlevé II equation from a large class of difference equations. The main idea is to re-interpret singularity confinement in terms of the value distribution of meromorphic solutions of difference equations. Solutions with sufficiently many non-confined singularities are shown to satisfy an inequality of the form Lemma 2.2 and hence they have infinite order.

First-order difference equations

The existence of nontrivial meromorphic solutions of the first-order nonlinear difference equation y¯=R(y), where R is a rational function with constant coefficients, is well established. However, the complete treatment of Eq. (4.1) is scattered in a number of papers; see, for instance Kimura [30], Shimomura [31] and Yanagihara [26]. Here we present a straightforward proof of the existence of meromorphic solutions of (4.1) by introducing suitable contraction mappings in appropriate Banach

Discussion

In this paper we have shown that all non-rational meromorphic solutions of a class of difference equations are of infinite order. Out of the remaining equations within the class (1.1) the only one which may have finite-order meromorphic solutions having comparably many singularities of two types is the difference Painlevé II equation. This is in some way analogous to what happens with the (differential) Painlevé II equation, which also has meromorphic solutions with two types of singularities

Acknowledgement

The research reported in this paper was supported by EPSRC grant number GR/R92141/01.

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    Current address: University of Joensuu, Department of Mathematics, P. O. Box 111, FI-80101 Joensuu, Finland.

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