IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i4p580-d1339056.html
   My bibliography  Save this article

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Author

Listed:
  • Anton E. Kulagin

    (Division for Electronic Engineering, Tomsk Polytechnic University, 30 Lenina av., 634050 Tomsk, Russia)

  • Alexander V. Shapovalov

    (Department of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., 634050 Tomsk, Russia
    Laboratory for Theoretical Cosmology, International Centre of Gravity and Cosmos, Tomsk State University of Control Systems and Radioelectronics, 40 Lenina av., 634050 Tomsk, Russia)

Abstract

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

Suggested Citation

  • Anton E. Kulagin & Alexander V. Shapovalov, 2024. "A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term," Mathematics, MDPI, vol. 12(4), pages 1-22, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:580-:d:1339056
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/4/580/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/4/580/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. V. V. Belov & A. Yu. Trifonov & A. V. Shapovalov, 2002. "The trajectory-coherent approximation and the system of moments for the Hartree type equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 32, pages 1-46, January.
    2. Alexander V. Shapovalov & Anton E. Kulagin, 2021. "Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media," Mathematics, MDPI, vol. 9(23), pages 1-17, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hari Mohan Srivastava, 2022. "Higher Transcendental Functions and Their Multi-Disciplinary Applications," Mathematics, MDPI, vol. 10(24), pages 1-3, December.
    2. Anton E. Kulagin & Alexander V. Shapovalov, 2023. "Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain," Mathematics, MDPI, vol. 11(3), pages 1-18, January.
    3. Alexander V. Shapovalov & Anton E. Kulagin, 2021. "Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media," Mathematics, MDPI, vol. 9(23), pages 1-17, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:580-:d:1339056. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.