A Discrete Inverse Scattering Transform for Q3$_delta$

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.

💡 Research Summary

The paper presents a fully discrete inverse scattering transform (IST) for solving the initial‑value problem of the Q3(\delta) lattice equation, a member of the ABS classification of integrable partial difference equations. The authors first define the admissible initial data: a real‑valued function prescribed on an infinite “staircase” subset of an N‑dimensional integer lattice (\mathbb{Z}^N). The data must satisfy a summability condition (\sum{k\in\mathbb{Z}}|u_{k+1}-u_k|<\infty), which guarantees that the forward scattering problem can be reduced to a one‑dimensional discrete spectral problem.

In the forward scattering stage, a Lax pair is employed to construct a discrete spatial operator. Jost solutions (\phi_n(\lambda)) and (\psi_n(\lambda)) are defined for a complex spectral parameter (\lambda). Their analytic properties split the spectrum into a continuous part (real (\lambda)) and a discrete part (complex eigenvalues corresponding to bound states). The scattering matrix (T(\lambda)) yields the reflection coefficient (r(\lambda)), transmission coefficient (t(\lambda)), and a set of discrete eigenvalues ({\lambda_j}) together with norming constants ({c_j}). The authors give explicit formulas for these quantities and prove their existence under the imposed summability condition.

The inverse problem is formulated via a discrete Gelfand‑Levitan‑Marchenko (GLM) equation. The kernel (K_{n,m}) satisfies
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