Discrete Integrable Equations over Finite Fields

TL;DR

This paper explores discrete integrable equations over finite fields, resolving indeterminacies via rational function fields, and presents explicit soliton solutions and their periods, linking to the singularity confinement method.

Contribution

It introduces a method to handle indeterminacies of discrete integrable equations over finite fields using rational functions and provides explicit soliton solutions.

Findings

Explicit soliton solutions over finite fields are derived.

The periods of soliton solutions are determined.

The approach relates to the singularity confinement criterion.

Abstract

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.