Soliton Solutions for ABS Lattice Equations: I Cauchy Matrix Approach

TL;DR

This paper reviews the construction of soliton solutions for ABS lattice equations, focusing on KdV type equations, and introduces a method to derive N-soliton solutions for most equations in the ABS classification.

Contribution

It develops a Cauchy matrix approach to construct explicit N-soliton solutions for ABS lattice equations, excluding the elliptic case Q4.

Findings

Explicit N-soliton solutions for KdV type lattice equations

Extension of soliton construction to all ABS equations except Q4

Enhanced understanding of integrability in discrete lattice systems

Abstract

In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations "of KdV type" that were known since the late 1970s and early 1980s. In this paper we review the construction of soliton solutions for the KdV type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.