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Integrable Nonlocal PT Symmetric and Reverse Space-Time Nonlinear Schrödinger Equations

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Parity-time Symmetry and Its Applications

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 280))

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Abstract

In this chapter a review of recent advances related to the emerging field of integrable nonlocal nonlinear PT symmetric, reverse space-time and reverse time only equations is presented. Starting from the well-known AKNS theory, it is shown how to obtain a host of nonlocal integrable equations previously discovered by the authors. Included are the nonlocal PT symmetric (1 + 1)D nonlinear Schrödinger (NLS) equation and its multi-component generalization; the reverse space-time and reverse time only NLS equations along with their vector versions. The inverse scattering transform associated with the nonlocal NLS hierarchy is briefly summarized and one soliton solutions corresponding to each of the above case are presented. The discrete nonlocal PT symmetric, reverse space-time and reverse time only NLS equations are also discussed. Starting from the Ablowitz-Ladik scattering problem, it is shown that all these discrete models arise from simple symmetry reductions.

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Acknowledgements

MJA was partially supported by NSF under Grant No. DMS-1712793.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado, Boulder, CO, USA

    Mark J. Ablowitz

  2. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Ziad H. Musslimani

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  1. Mark J. Ablowitz
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  2. Ziad H. Musslimani
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Correspondence to Ziad H. Musslimani .

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Editors and Affiliations

  1. College of Optics and Photonics, University of Central Florida, Orlando, FL, USA

    Demetrios Christodoulides

  2. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

    Jianke Yang

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Ablowitz, M.J., Musslimani, Z.H. (2018). Integrable Nonlocal PT Symmetric and Reverse Space-Time Nonlinear Schrödinger Equations. In: Christodoulides, D., Yang, J. (eds) Parity-time Symmetry and Its Applications. Springer Tracts in Modern Physics, vol 280. Springer, Singapore. https://doi.org/10.1007/978-981-13-1247-2_17

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  • DOI: https://doi.org/10.1007/978-981-13-1247-2_17

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  • Print ISBN: 978-981-13-1246-5

  • Online ISBN: 978-981-13-1247-2

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