Solutions to an autonomous discrete KdV equation via Painlev\'e-type ordinary difference equations

Abstract

Hirota's discrete KdV equation is a well-known integrable two-dimensional partial difference equation regarded as a discrete analogue of the KdV equation. In this paper, we show that a variation of Hirota's discrete KdV equation with an additional parameter admits two types of exact solutions: discrete Painlev\'e transcendent solutions and periodic solutions described by Painlev\'e-type ordinary difference equations.