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Rogue waves in the nonlocal \({\mathcal {PT}}\)-symmetric nonlinear Schrödinger equation

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Abstract

Rogue waves in the nonlocal \({\mathcal {PT}}\)-symmetric nonlinear Schrödinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only \(n(n+1)\), but also \(n(n-1)+1\) and \(n^2\), where n is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time or develop collapsing singularities, depending on their types as well as values of their free parameters. In addition, the solution dynamics exhibits rich patterns, most of which have no counterparts in the local NLS equation.

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Acknowledgements

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-12-1-0244 and the National Science Foundation under award number DMS-1616122. The work of B.Y. is supported by a visiting-student scholarship from the Chinese Scholarship Council.

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  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Bo Yang

  2. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, 05401, USA

    Jianke Yang

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Yang, B., Yang, J. Rogue waves in the nonlocal \({\mathcal {PT}}\)-symmetric nonlinear Schrödinger equation. Lett Math Phys 109, 945–973 (2019). https://doi.org/10.1007/s11005-018-1133-5

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