Discrete Painleve equations and discrete KdV equation over finite fields

TL;DR

This paper reviews recent advances in discrete integrable equations over finite fields, focusing on discrete Painleve equations and the discrete KdV equation, including their well-definedness, solutions, and properties like almost good reduction.

Contribution

It provides a comprehensive review of methods to define and analyze discrete Painleve and KdV equations over finite fields, introducing concepts like almost good reduction and solution construction.

Findings

Discrete Painleve equations are well-defined over finite fields via space of initial conditions.

The property of almost good reduction helps avoid indeterminacy in finite field equations.

Explicit soliton solutions and periods are obtained for the discrete KdV equation over finite fields.

Abstract

We investigate some of the discrete Painleve equations (dPII, qPI and qPII) and the discrete KdV equation over finite fields. The first part concerns the discrete Painleve equations. We review some of the ideas introduced in our previous papers and give some detailed discussions. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. We then extend them to the field of p-adic numbers and observe that they have a property that is called an `almost good reduction' of dynamical systems over finite fields. We can use this property, which can be interpreted as an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero. In the second part we study the discrete KdV equation. We review the previous discussions and present a way to resolve the indeterminacy of the equation by treating it over a field of rational functions instead of the finite field itself. Explicit forms of soliton solutions and their periods over finite fields are obtained. Note: This is a review article on the recent developments in the theory of discrete integrable equations over finite fields based on arXiv:1201.5429, arXiv:1206.4456, arXiv:1209.0223. This article is published as the proceedings of the domestic conference "The breadth and depth of nonlinear discrete integrable systems" in RIMS, Kyoto University, Japan, on August 2012.