VOLUME 67, NUMBER 14 PHYSICAL REVIEW LETTERS Discrete Versions of the Painleve Equations 30 SEPTEMBER 1991 A. Ramani Centre de Physique Theorique, Ecole Polytechnique, 91/28 Palaiseau, France B. Grammaticos Laboratoire de Physique NucIeaire, Uni versite de Paris VII, Tour 24-14, 75251 Paris CEDEX 05, France 3. Hietarinta Department of Physics, University of Turku, 20500 Turku, Finland (Received 10 April 1991) We present discrete forms of the Painleve transcendental equations P[[i, P]v, and Pv that complement the list of the already known P] and P[i. These, most likely integrable, nonautonomous mappings go over to the usual Painleve equations in the continuous limit, while in the autonomous limit we recover discrete systems that belong to the integrable family of Quispel et al. Finally, we show that the discrete Painleve mappings satisfy the same reduction relations as the continuous Painleve transcendents, name- ly, Pv [Ptii, Piv] PACS numbers: 05.50.+q, 02.90.+p Painleve transcendents occur frequently in physical models. The Ising model is perhaps the best known among them but these transcendents also appear in several other statistical-mechanics models related to con- formal field theory [1]. Another well-known application domain of the Painleve equations is that of integrable partial differential equations (PDE's) [2]. Reductions of integrable PDE's (and sometimes also of nonintegrable ones) often lead to one of these transcendental equations, a fact that makes possible the formulation of special solu- tions for the equation at hand. It is in these domains that the first two "discrete" transcendents have made their ap- pearance [3]. The aim of this paper is to derive the forms of the next three discrete Painleve equations using the new method of singularity confinement [4]. The (continuous) Painleve equations were discovered at the beginning of the century by Painleve and Gambier [5]. The method used for the derivation of these tran- scendental equations is related to what came to be known in the past decade as singularity analysis [6]. These equations have the so-called Painleve property, i.e. , their solutions are meromorphic functions of the independent variable, or, equivalently, their (movable) singularities are just poles [7]. Their solutions were given only in the past few years. Following the pioneering work of Ablowitz and Segur [8], it was shown that the Painleve equations can be linearized in terms of integro-dif- ferential equations, using the inverse scattering transform scheme. Recently the inverse monodromy (isomonodro- my) method has been developed for the study of the Pain- leve equations [9]. Another feature suggesting nice be- havior is that the Painleve equations can be written in the Hirota bilinear form [10]. The question of the existence of a discretized form of the Painleve equations arose naturally, given the intense activity around discrete systems. While mappings were initially used as prototypes for the study of chaos, the re- cent trends are towards the complementary direction, x„(an+ b) + c &n+]+Xn — I 1 — x„ (2) has also been obtained [13] in a way closely parallel to the one used for PDE's, as a simi', arity reduction of the discrete version of the modified Korteweg-de Vries equa- tion. The derivation of the discrete forms of Painleve equa- tions has so far been fortuitous, because there was no discrete analog of the singularity analysis method. Quite recently a "singularity confinement" method [4] was pro- posed relating the integrable character of discrete systems to their singularity structure. We now have, for map- pings, the equivalent of the Ablowitz-Ram ani-Segur (ARS) conjecture [14] for partial and ordinary differ- ential equations (ODE's). The implementation of the singularity confinement method is quite simple. Given a mapping, one must first find all possible ways a singularity can emerge (this step follows closely the first step of the algorithm for ODE's where one looks for all possible leading singular behav- iors). The system is said to have passed the test (and is thus a candidate for integrability) if this divergence does not propagate in (discrete) time, i.e. , that it remains confined. The second step is therefore to find how far it that of integrability. Numerous studies have been pub- lished concerning the construction of integrable mappings and lattices (higher-dimensional mappings) [11]. Several mappings that naturally appear in physical applications have turned out to be discrete analogs of the Painleve equations. For example, the computation of a certain partition function in a model of 2D quantum gravity led to the discrete form d P~ of P~ -[3]: x. + ~+x. — ~+x„=(an+&)/x„+c. Its solution using the isomonodromy approach has been given in [12]. A discrete form d-P~~ of Pit, Oc 1991 The American Physical Society 1829