Rules of discretization for Painlev\'e equations

TL;DR

This paper defines the discrete Painlevé property, establishes discretization rules to preserve it, and introduces new methods for testing and finding discrete Lax pairs, advancing the understanding of discrete integrable systems.

Contribution

It provides a precise definition of the discrete Painlevé property, new discretization rules, and methods for testing and constructing discrete Lax pairs, enhancing the analysis of discrete integrable equations.

Findings

Defined the discrete Painlevé property.

Introduced a new method to generate no-log conditions.

Provided a direct method to search for discrete Lax pairs.

Abstract

The discrete Painlev\'e property is precisely defined, and basic discretization rules to preserve it are stated. The discrete Painlev\'e test is enriched with a new method which perturbs the continuum limit and generates infinitely many no-log conditions. A general, direct method is provided to search for discrete Lax pairs.