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Exactly Solvable Wadati Potentials in the PT-Symmetric Gross-Pitaevskii Equation

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Non-Hermitian Hamiltonians in Quantum Physics

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 184))

Abstract

This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: \(V=-W^2+iW_x\). We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.

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Notes

  1. 1.

    Yet these have not been unheard of before. For instance, the potentials (3) appear in the context of supersymmetry [14–16] and have applications in subatomic physics (where they were utilized for the modeling of neutrino oscillations [17]).

  2. 2.

    In nonlinear dynamics, the bubble refers to a particular class of nontopological solitons with nontrivial boundary conditions [28, 29]. In contrast to the strict mathematical terminology, we use this word in a broad physical sense here—as a synonym of a hole in the constant-density condensate. The optical equivalent of the condensate bubble is dark soliton.

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Acknowledgments

This work was supported by the NRF of South Africa (grants UID 85751, 86991, and 87814) and the FCT (Portugal) through the grants UID/FIS/00618/2013 and PTDC/FIS-OPT/1918/2012. One of the authors (IVB) also thanks the Israel Institute for Advanced Studies for partial financial support.

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Authors and Affiliations

  1. National Institute for Theoretical Physics, Stellenbosch, Western Cape, South Africa

    I. V. Barashenkov

  2. Department of Mathematics, University of Cape Town, Rondebosch, Cape Town, 7701, South Africa

    I. V. Barashenkov

  3. Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C8, 1749-016, Lisboa, Portugal

    D. A. Zezyulin & V. V. Konotop

Authors
  1. I. V. Barashenkov
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  2. D. A. Zezyulin
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  3. V. V. Konotop
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Correspondence to I. V. Barashenkov .

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Editors and Affiliations

  1. Dipartimento di Energia, Università degli Studi di Palermo, Palermo, Italy

    Fabio Bagarello

  2. Dipartimento di Fisica e Chimica, Università degli Studi di Palermo, Palermo, Palermo, Italy

    Roberto Passante

  3. Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy

    Camillo Trapani

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Barashenkov, I.V., Zezyulin, D.A., Konotop, V.V. (2016). Exactly Solvable Wadati Potentials in the PT-Symmetric Gross-Pitaevskii Equation. In: Bagarello, F., Passante, R., Trapani, C. (eds) Non-Hermitian Hamiltonians in Quantum Physics. Springer Proceedings in Physics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-31356-6_9

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