Soliton solutions for Q3

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 $(Q3)_{δ=0}$ 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 $(Q3)_{δ=0}$. 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.

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