Nonlocal reductions of the Ablowitz-Ladik equation

The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and th...

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Bibliographic Details
Published in:arXiv.org 2017-11
Main Authors: Grahovski, Georgi G, Mohammed, Amal J, Susanto, Hadi
Format: Article
Language:English
Subjects:
Eigenvectors
Fourier transforms
Inverse scattering
Mathematical analysis
Nonlinear equations
Schrodinger equation
Online Access:Get full text
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Summary:The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as 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, the paper studies the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness relation for the associated Jost solutions is proved. Based on this, the expansion formula over the complete set of Jost solutions is derived. This will allow one to interpret the inverse scattering transform as a generalised Fourier transform.
ISSN:2331-8422
DOI:10.48550/arxiv.1711.08419