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

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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 13 scientific papers (total in 13 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 (13)

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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 13 articles:

Ya-Hui Liu, Rui Guo, Jian-Wen Zhang, “Inverse scattering transform for the discrete nonlocal PT symmetric nonlinear Schrödinger equation with nonzero boundary conditions”, Physica D: Nonlinear Phenomena, 472 (2025), 134528
Y. Jiang, X. Wang, “Vector rogue waves and their dynamics in the nonlocal three-component Manakov system”, Phys. Scr., 98:12 (2023), 125235
J. I. Mustafa, K. S. Chiu, “Employing a modified Sumudu with a modified iteration method to solve the system of nonlinear partial differential equations”, Computational and Mathematical Methods, 2023 (2023), 6649037
H. Hamid, J. Mustafa, “The essential conditions for soliton solution of the non-local Manakov system”, 2nd International Conference of Mathematics, Applied Sciences, Information and Communication Technology, AIP Conf. Proc., 2834, no. 1, 2023, 080076
S. J. A. Malham, “Integrability of local and non-local non-commutative fourth-order quintic non-linear Schrodinger equations”, IMA J. Appl. Math., 87:2 (2022), 231–259
Y. Shen, B. Tian, T.-Y. Zhou, X.-T. Gao, “Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, $N$-fold Darboux transformation and explicit exact solutions”, Chaos, Solitons & Fractals, 164 (2022), 112460
X. Wang, Ch. Li, “Solitons, breathers and rogue waves in the coupled nonlocal reverse-time nonlinear Schrödinger equations”, Journal of Geometry and Physics, 180 (2022), 104619
A. Doikou, S. J. A. Malham, I. Stylianidis, “Grassmannian flows and applications to non-commutative non-local and local integrable systems”, Physica D, 415 (2021), 132744
J. Chen, B.-F. Feng, Y. Jin, “Gram determinant solutions to nonlocal integrable discrete nonlinear Schrodinger equations via the pair reduction”, Wave Motion, 93 (2020), 102487
Chen J., Yan Q., “Bright Soliton Solutions to a Nonlocal Nonlinear Schrodinger Equation of Reverse-Time Type”, Nonlinear Dyn., 100:3 (2020), 2807–2816
S. Y. Lou, “Multi-place physics and multi-place nonlocal systems”, Commun. Theor. Phys., 72:5 (2020), UNSP 057001
M. J. Ablowitz, X.-D. Luo, Z. H. Musslimani, “Discrete nonlocal nonlinear Schrodinger systems: integrability, inverse scattering and solitons”, Nonlinearity, 33:7 (2020), 3653–3707
V. S. Gerdjikov, “On the integrability of ablowitz-ladik models with local and nonlocal reductions”, Vii International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012015

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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