On the formulation of the NQC variable

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 and the eigenfunction approach 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.