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

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

Journal

Mathematics
, E-ISSN: 2227-7390

Output data

Year: 2023, Volume: 11, Number: 21, Article number : 4458, Pages count : 31 DOI: 10.3390/math11214458

Tags

Gelfand–Levitan–Krein–Marchenko equation; inverse coefficient problem; inverse scattering problem

Authors

Kabanikhin S.I. ^1,2,3 , Shishlenin Maxim A. ^1,2,3 , Novikov Nikita S. ^1,2,3 , Prokhoshin Nikita ³

Affiliations

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

Funding (1)

Министерство науки и высшего образования РФ
Mathematical Center in Akademgorodok

075-15-2019-1613, 075-15-2022-281

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.

Kabanikhin S.I. , Shishlenin M.A. , Novikov N.S. , Prokhoshin N.
Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations
Mathematics. 2023. V.11. N21. 4458 :1-31. DOI: 10.3390/math11214458 WOS Scopus РИНЦ OpenAlex

Submitted:	Jun 25, 2023
Accepted:	Oct 16, 2023
Published print:	Oct 27, 2023
Published online:	Oct 27, 2023

Web of science:	WOS:001100201200001
Scopus:	2-s2.0-85176336017
Elibrary:	63906171
OpenAlex:	W4387971124

DB	Citing
OpenAlex	2
Scopus	2
Web of science	2