Soliton solutions for Q3

TL;DR

This paper constructs N-soliton solutions for the Q3 equation from the Adler-Bobenko-Suris classification, utilizing its relationship with the NQC equation and tau-functions of the Hirota-Miwa equation.

Contribution

It introduces a novel method to derive N-soliton solutions for Q3 by linking it to the NQC equation and tau-functions, expanding the solution space.

Findings

Derived explicit N-soliton solutions for Q3.

Established a 4-to-1 relationship between NQC and Q3.

Connected solutions to tau-functions of the Hirota-Miwa equation.

Abstract

We construct N-soliton solutions to the equation called Q3 in the recent Adler-Bobenko-Suris classification. An essential ingredient in the construction is the relationship of $(Q3)_{\delta=0}$ to the equation proposed by Nijhoff, Quispel and Capel in 1983 (the NQC equation). This latter equation has two extra parameters, and depending on their sign choices we get a 4-to-1 relationship from NQC to $(Q3)_{\delta=0}$. This leads to a four-term background solution, and then to a 1-soliton solution using a Backlund transformation. Using the 1SS as a guide allows us to get the N-soliton solution in terms of the tau-function of the Hirota-Miwa equation.