A Lax pair for the complete QRT mapping

TL;DR

This paper explicitly constructs 2x2 Lax pairs for the entire 18-parameter family of QRT maps, which are transformations on elliptic curves, enhancing understanding of their integrability.

Contribution

The paper provides the first explicit Lax pairs for all QRT maps within the 18-parameter family, broadening the scope of integrability analysis.

Findings

Explicit 2x2 Lax pairs for all QRT maps derived

Enhanced understanding of the integrability of QRT maps

Framework applicable to a broad class of elliptic curve transformations

Abstract

QRT maps are translations on smooth biquadratic curves, also known as elliptic curves. Special cases of QRT maps are known to arise as compatibility conditions for an associated system of linear equations, known as a Lax pair. Here, we provide $2\times 2$ Lax pairs explicitly for the whole 18-parameter family of QRT maps.