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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
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
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