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Coupled Nonlinear Schrödinger Equations with Gain and Loss: Modeling \(\mathcal {PT}\) Symmetry

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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

Coupled nonlinear Schrödinger (NLS) equations is an ubiquitous model describing wave propagation in diverse physical systems. In combination with gain and loss exactly balanced with each other, this model allows for modeling parity (\(\mathcal {P}\)) and time (\(\mathcal {T}\)) symmetries in frameworks beyond the non-Hermitian quantum mechanics, where they have been introduced originally. Being open, i.e. not conserving energy, such systems nevertheless bear many properties which are characteristic for conservative models. This allows one to explore various wave phenomena in \(\mathcal {PT}\)-symmetric settings, including bright and dark solitons and their interactions with defects, soliton switches, resonant wave interactions, wave collapse, etc. In this Chapter an overview of some recent results on these processes is presented. The outcomes are interpreted in contexts of nonlinear optics and matter wave theory.

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Acknowledgements

Author is grateful to D. A. Zezyulin, B. A. Malomed, F. Kh. Abdullaev, Y. V. Bludov, G. Huang, C. Hang, Y. V. Kartashov, M. Trippenbah, M. Ögren, R. Driben, D. E. Pelinovsky, J.-P. Dias, M. Figueira, T. Wasak, and P. Szańkowski, for fruitful collaboration on the results reported in this chapter.

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  1. Departamento de Física and Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

    Vladimir V. Konotop

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  1. Vladimir V. Konotop
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Correspondence to Vladimir V. Konotop .

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  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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Konotop, V.V. (2018). Coupled Nonlinear Schrödinger Equations with Gain and Loss: Modeling \(\mathcal {PT}\) Symmetry. 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_14

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  • DOI: https://doi.org/10.1007/978-981-13-1247-2_14

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