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This paper develops a complete inverse scattering method for a discrete, parity‑time (PT) symmetric, nonlocal nonlinear Schrödinger equation when the solution does not decay at infinity but instead approaches a nonzero constant of arbitrarily large size. In plain terms, the authors show how to analyze and reconstruct solutions on an infinite lattice when each end of the lattice sits on a nonzero background that can be large. That situation breaks the assumptions used in earlier scattering theories and so required new tools.
The core technical work is to set up and solve the direct and inverse scattering problems for the discrete equation. The authors write the equation in a Lax pair form (a pair of linear equations whose compatibility encodes the nonlinear dynamics), introduce a uniformization variable to remove square‑root branchings that appear for large backgrounds, and then study the eigenfunctions and scattering data. They prove results about analyticity (where functions are complex differentiable), symmetry relations, and asymptotic behavior of those objects. The inverse problem is solved by formulating and solving a Riemann–Hilbert problem, which gives a reconstruction formula for the lattice potential (the field values) even when an arbitrary finite number of simple discrete eigenvalues are present.
One striking outcome is the identification of two kinds of localized waves (solitons) that appear in the focusing version of the model under large nonzero backgrounds. The first is an “oscillating soliton,” associated with an eigenvalue lying on the unit circle in the spectral plane; these can look like oscillatory dark or anti‑dark structures along the lattice sites. The second is a “breather,” coming from a symmetric pair of eigenvalues around the unit circle and showing an internal periodic oscillation in time. According to the authors, the oscillating soliton has not been reported before for this PT‑symmetric nonlocal discrete model or the related Ablowitz–Ladik equation, and the breather does not occur when the background amplitudes are small.
The paper also gives explicit determinant formulas for N‑soliton solutions with arbitrarily large nonzero boundary amplitudes. Using those formulas the authors describe multi‑soliton interactions, including collisions among oscillating dark and anti‑dark solitons and combinations of an oscillating soliton with a breather. These are concrete, exact solutions that illustrate the new nonlinear phenomena that large backgrounds permit.
Important caveats: the work focuses on the case where a particular phase condition holds (expressed in the paper as σ e^{iΔθ} = 1) and on backgrounds with amplitude squared greater than 1; other parameter regimes require different treatment. The analysis is mathematical and technical: it assumes the discrete eigenvalues are simple and builds on the Riemann–Hilbert framework. The results described are derived from this analytical framework; the paper does not claim experimental realization or numerical exploration beyond the exact solutions it constructs.