2017/11/22 by G. G. Grahovski, Georgi G. Grahovski, Amal J. Mohammed +3
Mathematics · Physics and Astronomy · #Completeness (order theory) #Eigenfunction #Fourier transform #Inverse #Inverse scattering problem #Inverse scattering transform #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum inverse scattering method #Scattering #Scattering theory #math-ph #math.MP #nlin.PS #nlin.SI
paper · pdf · doi:10.1134/s0040577918100021
published as Theor. Math. Phys. 197 (2018), 1412-1429 · 20 pages, 1 (png) figure, LaTeX
arxiv created 2017/11/22 · openalex created_date 2017/12/04 · openalex publication_date 2018/10/01 · arxiv updated 2019/10/15 · openalex updated_date 2026/08/05
Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schrödinger equation (called the Ablowitz–Ladik equation) with PT symmetry. This includes the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data, and the fundamental analytic solutions. In addition, we study the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann–Hilbert problem, we derive the one- and two-soliton solutions for the nonlocal Ablowitz–Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the complete set of Jost solutions. This allows interpreting the inverse scattering transform as a generalized Fourier transform.