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

Sergey Kabanikhin, Maxim Shishlenin (), Nikita Novikov () and Nikita Prokhoshin
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Sergey Kabanikhin: Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
Maxim Shishlenin: Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
Nikita Novikov: Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, 630090 Novosibirsk, Russia
Nikita Prokhoshin: Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia

Mathematics, 2023, vol. 11, issue 21, 1-31

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.

Keywords: Gelfand–Levitan–Krein–Marchenko equation; inverse coefficient problem; inverse scattering problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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