A constructive approach to the soliton solutions of integrable quadrilateral lattice equations

TL;DR

This paper develops a constructive method for deriving N-soliton solutions of integrable quadrilateral lattice equations, specifically applying it to Adler's Q4 equation, by linking superposition principles and Riccati map solutions.

Contribution

It introduces a novel approach combining superposition and Riccati maps to explicitly construct soliton solutions for multidimensionally consistent lattice equations.

Findings

Derived explicit N-soliton solutions for Adler's Q4 equation

Established a connection between superposition principles and Riccati maps

Provided a general framework for soliton solutions in integrable lattice equations

Abstract

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Backlund transformation, and the method of solving a Riccati map by exploiting two kn own particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler's lattice equation (or Q4).