Elliptic Solutions of ABS Lattice Equations

TL;DR

This paper constructs elliptic N-soliton solutions for all ABS lattice equations except Q4, introduces an elliptic Cauchy matrix approach, and reveals a new Lax representation for Q3, advancing the understanding of integrable lattice equations.

Contribution

It provides a unified construction of elliptic solutions for the ABS list using an elliptic Cauchy matrix and introduces a novel Lax representation for Q3.

Findings

Constructed elliptic N-soliton solutions for ABS equations (except Q4)

Developed an elliptic Cauchy matrix approach for solution construction

Discovered a new Lax representation for the Q3 equation

Abstract

Elliptic N-soliton-type solutions, i.e. solutions emerging from the application of N consecutive B\"acklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is treated elsewhere. The main construction, which is based on an elliptic Cauchy matrix, is performed for the equation Q3, and by coalescence on certain auxiliary parameters, the corresponding solutions of the remaining equations in the list are obtained. Furthermore, the underlying linear structure of the equations is exhibited, leading, in particular, to a novel Lax representation of the Q3 equation.