Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps

TL;DR

This paper develops an algebro-geometric integration method for the Q1 lattice equation using nonlinear integrable symplectic maps and Riemann theta functions, advancing the understanding of discrete integrable systems.

Contribution

It introduces a novel Lax pair for Q1, nonlinearises it into integrable symplectic maps, and derives a theta function solution for the lattice equation.

Findings

Derived a new Lax pair for Q1.

Established integrable symplectic maps from the Lax pair.

Obtained a Riemann theta function expression for solutions.

Abstract

The Q1 lattice equation, a member in the Adler-Bobenko-Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearised to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker-Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps.