Integrable maps from Galois differential algebras, Borel transforms and number sequences

TL;DR

This paper introduces a new class of integrable lattice maps derived from polynomial dynamical systems, utilizing Galois differential algebras, Borel regularization, and category theory, revealing links between number sequences and integrability.

Contribution

It presents a novel discretization method preserving symmetries and integrability using Galois differential algebras and category theory, and explores connections to number sequences.

Findings

New integrable lattice maps derived from polynomial systems

Discretization preserves symmetries and integrability

Connection established between number sequences and integrability

Abstract

A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability [40]. Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.