A Discrete Inverse Scattering Transform for Q3$_δ$

We derive a fully discrete Inverse Scattering Transform as a method for solving the initial-value problem for the Q3$_δ$ lattice (difference-difference) equation for real-valued solutions. The initial condition is given on an infinite staircase within an N-dimensional lattice and must obey a given summability condition. The forward scattering problem is one-dimensional and the solution to Q3$_δ$ is expressed through the solution of a singular integral equation. The solutions obtained depend on N discrete independent variables and N parameters.

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