Soliton solutions for Q3

We construct N-soliton solutions to the equation called Q3 in the recent Adler‐Bobenko‐Surisclassification.Anessentialingredientintheconstruction 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 B¨ acklund transformation. Using the 1SS as a guide allows us to get the N-soliton solution in terms of the τ-function of the Hirota‐Miwa equation.

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