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Nonlocal Reductions of the Ablowitz–Ladik Equation

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Essex, Colchester, UK

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

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  1. G. G. Grahovski
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  2. A. J. Mohammed
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  3. H. Susanto
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Correspondence to G. G. Grahovski.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 197, No. 1, pp. 24–44, October, 2018.

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Grahovski, G.G., Mohammed, A.J. & Susanto, H. Nonlocal Reductions of the Ablowitz–Ladik Equation. Theor Math Phys 197, 1412–1429 (2018). https://doi.org/10.1134/S0040577918100021

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