On the formulation of the NQC variable
Leilei Shi, Cheng Zhang, Da-jun Zhang
TL;DR
This work develops a unified framework for the NQC variable $S(a,b)$ by linking the Cauchy matrix formulation, Dbar methods, and spectral-eigenfunction approaches. A new single-eigenfunction formulation is introduced using a Dbar problem tied to the lattice non-potential mKdV Lax pair, from which the NQC equation is derived and $S(a,b)$ is explicitly characterized as $S(a,b)=\mathbf{r}^T(\mathbf{K}+a\mathbf{I})^{-1}(\mathbf{I}+\mathbf{M})^{-1}(\mathbf{K}+b\mathbf{I})^{-1}\mathbf{c}$. This result recovers the classical Cauchy matrix variables $S^{(i,j)}$ via suitable expansions and shows that all master functions can be generated from the single $S(a,b)$, thereby unifying multiple perspectives (Cauchy matrix, Dbar, and Lax-pair eigenfunctions) for the NQC hierarchy. The paper also clarifies connections to the lpKdV and lattice Schwarzian KdV equations through Miura-type relations and degenerations, and constructs $N$-soliton solutions within this framework, highlighting the practical impact for integrable lattice systems.
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.