A Revisit to the ABS H2 Equation

TL;DR

This paper revisits the ABS H2 equation, exploring its relation to the Cauchy matrix scheme, its 3D consistency, Lax pair, bilinear form, continuum limits, and its connection to the KdV eigenfunction.

Contribution

It provides a detailed analysis of the H2 equation's integrability properties and reveals new relations to lattice equations and the KdV eigenfunction.

Findings

H2 equation is linearly related to variables in the Cauchy matrix scheme.

The coupled system exhibits 3D consistency, Lax pair, and bilinear form.

S^{(1,0)} satisfies a 9-point lattice equation and relates to KdV eigenfunction in continuum limit.

Abstract

In this paper we revisit the Adler-Bobenko-Suris H2 equation. The H2 equation is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and $S^{(1,0)}$ in terms of their 3-dimensional consistency, Lax pair, bilinear form and continuum limits. It is shown that $S^{(1,0)}$ itself satisfies a 9-point lattice equation and in continuum limit $S^{(1,0)}$ is related to the eigenfunction in the Lax pair of the Korteweg-de Vries equation.