On integrals of non-autonomous dynamical systems in finite characteristic

TL;DR

This paper constructs integrals of motion for non-autonomous Painlevé equations over finite fields, revealing invariance properties, connections to Riccati equations, and methods for algebraic solutions in positive characteristic.

Contribution

It introduces a difference Lax form approach to find integrals of Painlevé equations over finite characteristic fields, highlighting invariance and solution construction techniques.

Findings

Integrals can be normalized to be invariant under affine Weyl group actions.

Components of reducible fibers relate to Riccati equation reductions.

Method for constructing non-rational algebraic solutions in positive characteristic.

Abstract

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlev\'e equation and the difference second Painlev\'e equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.