Do integrable mappings have the Painlevé property?

Abstract

We present an integrability criterion for discrete-time systems that is the equivalent of the Painleve property for systems of a continuous variable. It is based on the observation that for integrable mappings the singularities that may appear are confined, i.e., they do not propagate indefinitely when one iterates the mapping. Using this novel criterion we show that there exists a family of nonautonomous integrable mappings which includes the discrete Painleve equations P\, recently derived in a model of twodimensional quantum gravity, and P\\, obtained as a similarity reduction of the integrable modified Korteweg-de Vries lattice. These systems possess Lax pairs, a well-known integrability feature.PACS numbers: 05.50.+q, 02.90.+pThe study of discrete-time integrable systems is currently the focus of an intense activity [1]. The systems considered are either lattices, i.e., partial difference equations where both the spatial and time variables have been discretized, or mappings, i.e., finite degree-offreedom systems in discrete time. Integrable lattices are particularly interesting as their various continuous limits generate entire hierarchies of integrable partial differential equations (PDE's). Integrability for these systems is often associated with the existence of a Lax representation, a Zakharov-Shabat linearization, but also deduced from the existence of a sufficient number of integrals of motion in involution [2]. The situation is reminiscent of the status of nonlinear evolution equations in the 1970s. More and more integrable systems were being constructed, but whenever a new nonlinear PDE appeared in a physical application its integrability could be surmised only at the cost of lengthy numerical investigations. The situation was dramatically modified with the advent of the Painleve criterion [3]. A study of the singularity structure of the equation at hand allowed one, in most cases, to make a safe prediction of its integrable (or not) character. This development became possible only after the production of a "critical mass" of integrable PDE's on which the conjecture relating integrability and singularity structure could be tested. This is roughly what is happening now with discrete-time systems. More and more integrable lattices and mappings are appearing in the literature [4] but no criterion for the integrability of a new system existed until now. So there has been no way to predict the integrability of mappings short of exhibiting Lax pairs or a set of commuting integrals, i.e., proving the integrability. The present work gives a new criterion for assessing the integrability of discrete-time systems. It is based on the study of the movable singularities of a mapping and so it is, in some way, the analog of the Painleve criterion for continuous-time systems.In order to introduce our method let us study the integrable lattice of potential-Korteweg-de Vries type, presented in [2,5], that we write here as x <+*-x fi\ + -L-/-.

Some integrable maps and their Hirota bilinear forms

“…. , whose ratios τ n+3 τ n /(τ n+2 τ n+1 ) produce (9). By considering bilinear relations with constant coefficients, the minimal relation is found to be of order q = 12, corresponding to the matrix M with entries M ij = τ 12+i−j τ i+j−2 .…”

“…The first relevant tool here is the singularity confinement test, which was introduced in [9] as a heuristic method for identifying discrete systems that may be integrable. In its original form, this method has the drawback that many systems with confined singularities have positive algebraic entropy [15], but recently singularity confinement has been refined to include information about deautonomization, which renders it a more effective tool [10,31,35].…”

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

“…We apply a singularity confinement (SC) test to these difference equations [30]. Equation (11) does not pass the SC test except the case in which Eq.…”

The Gambier mapping, revisited

“…The main tool for the investigation of the integrability of the Gambier mapping will be the singularity confinement criterion [5]. A first remark before implementing the singularity confinement algorithm is that the singularities of a Riccati mapping are automatically confined.…”