Lattice eigenfunction equations of KdV-type

Abstract

We develop lattice eigenfunction equations of the lattice KdV equation, which are equations obeyed by auxiliary functions, or eigenfunctions, of the Lax pair of the lattice KdV equation. These equations are three-dimensionally consistent quad-equations, that are closely related to lattice equations in the Adler-Bobenko-Suris (ABS) classification. The connection between the H3(δ), Q1(δ), Q2 and Q3(δ) equations in the ABS classification and the lattice eigenfunction equations is explicitly showed. In particular, we provide a natural interpretation of the δ term in those equations. This can be understood as “interactions” between the eigenfunctions. Other integrable properties of the eigenfunction equations, such as exact solutions, discrete zero curvature conditions are also provided. We believe that the approach presented in this paper can be used as a means to search for integrable lattice equations.