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Integrable Nonlocal Reductions

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Symmetries, Differential Equations and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 266))

Abstract

We present some nonlocal integrable systems by using the Ablowitz–Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy–Kulish equations by utilizing the nonlocal reduction to the Fordy–Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz–Musslimani nonlocal reduction can be extended to super integrable equations. We obtain new nonlocal equations namely nonlocal super NLS and nonlocal super mKdV equations.

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Acknowledgements

This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).

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Authors and Affiliations

  1. Faculty of Science, Department of Mathematics, Bilkent University, 06800, Ankara, Turkey

    Metin Gürses

  2. Faculty of Science, Department of Mathematics, Hacettepe University, 06800, Ankara, Turkey

    Aslı Pekcan

Authors
  1. Metin Gürses
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  2. Aslı Pekcan
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Correspondence to Metin Gürses .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA, USA

    Victor G. Kac

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Peter J. Olver

  3. Department of Mathematics and Statistics, Université de Montréal, Montréal, QC, Canada

    Pavel Winternitz

  4. Civil Engineering Dept, Div.of Mechanics, Istanbul Technical University, Civil Engineering Department, Div.of Mechanics, Maslak (Istanbul), Turkey

    Teoman Özer

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Gürses, M., Pekcan, A. (2018). Integrable Nonlocal Reductions. In: Kac, V., Olver, P., Winternitz, P., Özer, T. (eds) Symmetries, Differential Equations and Applications. Springer Proceedings in Mathematics & Statistics, vol 266. Springer, Cham. https://doi.org/10.1007/978-3-030-01376-9_2

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