On the formulation of the NQC variable

TL;DR

This paper reviews existing formulations of the NQC variable in lattice equations and introduces a novel approach using eigenfunctions of a Lax pair and a new Dbar problem.

Contribution

It provides a new perspective on formulating the NQC variable through eigenfunctions of a Lax pair and introduces a novel Dbar problem for derivation.

Findings

Multiple known formulations of NQC variable reviewed

A new formulation via eigenfunctions of a Lax pair proposed

Introduction of a new Dbar problem for derivation

Abstract

The Nijhoff-Quispel-Capel (NQC) equation is a general lattice quadrilateral equation presented in terms of a function $S(a,b)$ where $a$ and $b$ serve as extra parameters. It can be viewed as the counterpart of Q3 equation which is the second top equation in the Adler-Bobenko-Suris list. In this paper, we review some known formulations of the NQC variable $S(a,b)$, such as the Cauchy matrix approach, the eigenfunction approach and via a spectral Wronskian. We also present a new perspective to formulate $S(a,b)$ from the eigenfunctions of a Lax pair of the lattice (non-potential) modified Korteweg de Vries equation. A new Dbar problem is introduced and employed in the derivation.