2010
DOI: 10.1007/s00020-010-1843-2
|View full text |Cite
|
Sign up to set email alerts
|

Construction of the Solution of the Inverse Spectral Problem for a System Depending Rationally on the Spectral Parameter, Borg–Marchenko-Type Theorem and Sine-Gordon Equation

Abstract: Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg-Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed. An application to sine-Gordon equation in laboratory coordinates is given.Mathematics Subject Classification (2010). Primary 34B07, 34A55; Secondary 34B20, 35Q51.

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
3

Citation Types

0
4
0

Year Published

2010
2010
2022
2022

Publication Types

Select...
3
3
1

Relationship

2
5

Authors

Journals

citations
Cited by 7 publications
(4 citation statements)
references
References 42 publications
0
4
0
Order By: Relevance
“…Weyl functions (also called Weyl-Titchmarsh or M-functions) and their generalizations are an important and much used tool in the spectral theory of differential equations (see a necessarily small part of recent papers and books on this topic and various references therein [3-5, 10, 11, 14, 16-18, 20, 32, 37, 41, 49, 51]). Following the seminal work [27] and more general constructions in [45,49] (see also some references therein), one can use structured operators to solve inverse problems for Krein, Dirac, canonical, nonclassical and non-self-adjoint systems and their discrete analogues (see, for instance, [2,15,16,37,[40][41][42]). It was proved in [16,36,37,[40][41][42] that the kernels of these structured operators are connected with the Weyl functions via some kinds of Fourier transformations and can be recovered directly from the Weyl functions.…”
Section: Introductionmentioning
confidence: 99%
“…Weyl functions (also called Weyl-Titchmarsh or M-functions) and their generalizations are an important and much used tool in the spectral theory of differential equations (see a necessarily small part of recent papers and books on this topic and various references therein [3-5, 10, 11, 14, 16-18, 20, 32, 37, 41, 49, 51]). Following the seminal work [27] and more general constructions in [45,49] (see also some references therein), one can use structured operators to solve inverse problems for Krein, Dirac, canonical, nonclassical and non-self-adjoint systems and their discrete analogues (see, for instance, [2,15,16,37,[40][41][42]). It was proved in [16,36,37,[40][41][42] that the kernels of these structured operators are connected with the Weyl functions via some kinds of Fourier transformations and can be recovered directly from the Weyl functions.…”
Section: Introductionmentioning
confidence: 99%
“…The case of nonlinear dependence on the spectral parameter z is of essential interest and was studied, for instance, in [9,18,19,23]. A particular case of system (1.2), where m = 2, p = 1 and j = I 2 , was considered in [26,32]. Further in Section 3, we study important cases, where Darboux matrix may be constructed explicitly.…”
Section: Introductionmentioning
confidence: 99%
“…(See [25][26][27][28] and references therein for more details on the method of operator identities.) Somewhat different kinds of structured operators play the main role in solving inverse problems for skew-self-adjoint Dirac systems, for auxiliary systems in the case of N-wave equations and for certain systems depending rationally on the spectral parameter (see, e.g., [6,20,22,25]). In particular, the matrix function Φ ′ 1 in (1.6) (or, equivalently, in (1.9)) may be considered as a direct analog of the Krein's accelerant.…”
Section: Introductionmentioning
confidence: 99%