The discrete Painlevé II equation and the classical special functions

Summary

Abstract

Exact Solutions for the discrete Painlevé II Equation(dP_II) are constructed. It is shown that dP_II 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?