Discrete Painlev\'e transcendent solutions to the multiplicative type discrete KdV equations

TL;DR

This paper demonstrates that solutions to multiplicative discrete KdV equations can be expressed using discrete Painlevé transcendents, linking integrable difference equations with special functions.

Contribution

It establishes a novel connection between multiplicative discrete KdV equations and $q$-Painlevé equations of types $A_J^{(1)}$, providing explicit solutions.

Findings

Solutions of multiplicative discrete KdV are given by $q$-Painlevé equations.

Links between integrable difference equations and special functions are clarified.

Provides explicit forms of solutions for specific $q$-Painlevé types.

Abstract

Hirota's discrete KdV equation is an integrable partial difference equation on $\mathbb{Z}^2$, which approaches the Korteweg-de Vries (KdV) equation in a continuum limit. In this paper, we show that its multiplicative-discrete versions have the special solutions given by the solutions of $q$-Painlev\'e equations of types $A_J^{(1)}$ $(J=3,4,5,6)$.