Abstract

We study the rational solutions of the discrete version of Painleve's fourth equation (d-PIV). The solutions are generated by applying Schlesinger transformations on the seed solutions −2z and −1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.

Introduction

One important question in the study of discrete versions of continuous differential equations concerns the existence of corresponding special solutions. For continuous Painlevé equations rational and special function solutions are known, and in many cases even a rigorous classification has been done. If one proposes a discrete version of a Painlevé equation it is not enough that in some continuous limit the continuous Painlevé equation is obtained, but in addition the proposed equation should share some further properties of the original equation. One of these properties should be the equivalent of the Painlevé property, called “singularity confinement”. This has already been used to propose discrete forms of the Painlevé equations. Other structures of the continuous Painlevé equations that have been shown to exist for the discrete ones include their relationships by coalescence limits and the existence of Hirota forms for these equations. What is still largely an open question is the fate of the special solutions (rational, algebraic, special function) known for the continuous case.

Abstract

Exact Solutions for the discrete Painlevé II Equation(dPII) are constructed. It is shown that dPII admits three kinds of exact solutions with determinant structure, namely, discrete Airy function solutions, rational solutions, and so-called “molecular type” solution. These solutions are expressed by classical special functions.

Introduction

The discrete Painlevé equations are now attracting much attention. One reason may be due to the importance of the six Painlevé equations in the continuous systems: their solutions play a role of special functions in the theory of nonlinear integrable systems.

We encounter various special functions when we reduce linear partial differential equations to ordinary differential equation by separation of variable. For example, when we solve the Helmholtz equation in polar coordinates, we separate the variable and we get the Bessel functions in radius. When we consider nonlinear integrable systems, simple separation of variables does not work indeed, but it is possible to reduce them to ordinary differential equations by considering traveling wave solutions or similarity solutions. In this case, it is believed that any reduced ordinary differential equations have so-called the “Painlevé property” and usually we have one of the six types of Painlevé equations. We may expect that the discrete Painlevé equations plays a similar role in the discrete integrable systems. systems. Here, natural questions arise: Is it possible to regard their solutions as the nonlinear versions of discrete analogue of special functions? What kind of solutions do we get for them?