Nonlocal Reductions of the Ablowitz–Ladik Equation

G. G. Grahovski; A. J. Mohammed; H. Susanto

doi:10.1134/S0040577918100021

Nonlocal Reductions of the Ablowitz–Ladik Equation

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

Mathematics

University of Essex

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

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.

Original language	British English
Pages (from-to)	1412-1429
Number of pages	18
Journal	Theoretical and Mathematical Physics(Russian Federation)
Volume	197
Issue number	1
DOIs	https://doi.org/10.1134/S0040577918100021
State	Published - 1 Oct 2018

Keywords

integrable system
nonlocal reduction
PT symmetry
Riemann–Hilbert problem
soliton

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abstract = "Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schr{\"o}dinger 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.",

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AU - Susanto, H.

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AB - 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.

KW - integrable system

KW - nonlocal reduction

KW - PT symmetry

KW - Riemann–Hilbert problem

KW - soliton

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Nonlocal Reductions of the Ablowitz–Ladik Equation

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Keywords

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