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Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

Author

Listed:
  • Sergey Kabanikhin

    (Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
    Sobolev Institute of Mathematics SB RAS, Akad. Koptyug Avenue, 4, 630090 Novosibirsk, Russia
    Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia)

  • Maxim Shishlenin

    (Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
    Sobolev Institute of Mathematics SB RAS, Akad. Koptyug Avenue, 4, 630090 Novosibirsk, Russia
    Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia)

  • Nikita Novikov

    (Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
    Sobolev Institute of Mathematics SB RAS, Akad. Koptyug Avenue, 4, 630090 Novosibirsk, Russia
    Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia)

  • Nikita Prokhoshin

    (Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia)

Abstract

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.

Suggested Citation

  • Sergey Kabanikhin & Maxim Shishlenin & Nikita Novikov & Nikita Prokhoshin, 2023. "Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations," Mathematics, MDPI, vol. 11(21), pages 1-31, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4458-:d:1268961
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