A Discrete Inverse Scattering Transform for Q3$_\delta$

TL;DR

This paper develops a fully discrete inverse scattering transform method to solve the initial-value problem for the Q3$_\

Contribution

It introduces a novel discrete inverse scattering transform specifically designed for the Q3$_\delta$ lattice equation, extending integrability techniques to difference-difference equations.

Findings

Derived a discrete inverse scattering transform for Q3$_\delta$.

Expressed solutions via singular integral equations.

Applicable to initial conditions on an infinite staircase.

Abstract

We derive a fully discrete Inverse Scattering Transform as a method for solving the initial-value problem for the Q3$_\delta$ 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$_\delta$ is expressed through the solution of a singular integral equation. The solutions obtained depend on N discrete independent variables and N parameters.