On the Dbar method and direct linearization approach of the lattice KdV type equations

The purpose of this paper is to bridge the gap between the Dbar method and the direct linearization approach for the lattice Korteweg-de Vries (KdV) type equations. We develop the Dbar method to study some discrete integrable equations in the Adler-Bobenko-Suris list. A Dbar problem is considered to define the eigenfunctions of the Lax pair of the lattice potential KdV equation. We show how an extra parameter is introduced in this approach so that the lattice potential modified KdV equation and lattice Schwarzian KdV equation are derived. We also explain how the so-called spectral Wronskians make sense in constructing the H3$(δ)$, Q1$(δ)$ and Q3$(δ)$ equations. Explicit formulae of multi-soliton solutions are given for the derived equations, from which one can see the connections between the direct linearization variables ($S^{(i,j)}$ and $V(p)$) and the eigenfunctions and their expansions respectively at infinity and a finite point.