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Teoreticheskaya i Matematicheskaya Fizika, 2018, Volume 197, Number 1, Pages 24–44
DOI: https://doi.org/10.4213/tmf9506 (Mi tmf9506) 		 
This article is cited in 12 scientific papers (total in 12 papers)

Nonlocal reductions of the Ablowitz–Ladik equation

G. G. Grahovski, A. Mohammed, H. Susanto

Department of Mathematical Sciences, University of Essex, Colchester, UK
Full-text PDF (555 kB) Citations (12)
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DOI: https://doi.org/10.4213/tmf9506
Abstract: Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schrödinger equation (called the Ablowitz–Ladik equation) with $\mathcal{PT}$ 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.
Keywords: integrable system, soliton, PT symmetry, nonlocal reduction, Riemann–Hilbert problem.
Received: 07.11.2017
English version:
Theoretical and Mathematical Physics, 2018, Volume 197, Issue 1, Pages 1412–1429
DOI: https://doi.org/10.1134/S0040577918100021
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: G. G. Grahovski, A. Mohammed, H. Susanto, “Nonlocal reductions of the Ablowitz–Ladik equation”, TMF, 197:1 (2018), 24–44; Theoret. and Math. Phys., 197:1 (2018), 1412–1429
Citation in format AMSBIB
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    • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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