On the construction of elliptic solutions of integrable birational maps

TL;DR

This paper introduces a systematic method for deriving explicit elliptic function solutions of integrable birational maps, utilizing addition theorems and algebraic curve detection, demonstrated on discretized Volterra chains.

Contribution

The paper develops a novel, systematic approach combining classical elliptic function theorems and experimental algebraic curve detection for solving integrable birational maps.

Findings

Successfully applied methods to discretized Volterra chains with 3 and 4 particles.

Provided explicit elliptic solutions for specific integrable birational maps.

Enhanced understanding of the structure of solutions in integrable discrete systems.

Abstract

We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are: (i) application of classical addition theorems for elliptic functions, and (ii) experimental technique to detect an algebraic curve containing a given sequence of points in a plane. These methods are applied to Kahan-Hirota-Kimura discretizations of the periodic Volterra chains with 3 and 4 particles.