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Prog. Theor. Phys. Vol. 80 No. 5 (1988) pp. 774-792
The Marchenko Equation of the Hilbert-Schmidt Type for the N ×N Zakharov-Shabat Operator
Naruyoshi Asano and
Yusuke Kato
Laboratory of Applied Mathematics and Mathematical Physics
Department of Information Science, Faculty of Engineering, Utsunomiya
University, Utsunomiya 321
(Received May 6, 1988)
Abstract:
The Marchenko equation for the N ×N
Zakharov-Shabat operator on the line is given via the Riemann-Hilbert
problem. It is derived on the basis of the direct spectral problem which was
solved through the Fredholm integral equation. In the 2 ×2 problem
the Marchenko equations for the right and the left inverse problems are
separated but in the N ×N case we obtain a coupled Marchenko
equations. The integral kernel of the Marchenko equation is shown to be of
the Hilbert-Schmidt type for a general class of Schwarz type potentials and
hence in the inverse problem the equation is solvable by the theorem of the
Fredholm alternatives. Finally reduction to the 2 ×2 case is briefly
discussed.
URL :
http://ptp.ipap.jp/link?PTP/80/774/
DOI : 10.1143/PTP.80.774
References:
- Y. Kato and H. Nakajima, Prog. Theor. Phys. 76 (1987), 981[PTP].
- D. J. Kaup, Stud. Appl. Math. 55 (1976), 9.
- A. B. Shabat, Functional Anal. Appl. 9 (1975), 75.
- R. Beals and R. R. Coifman, Commun. Pure. Appl. Math. 37 (1984), 39.
-
T. Kawata, J. Phys. Soc. Jpn. 57 (1988), 422[JPSJ].
-
P. J. Caudrey, Physica 6D (1982), 51[Elsevier].
- N. Asano and Y. Kato, Prog. Theor. Phys. 74 (1985), 1005[PTP].
Citing Article(s) :
-
Progress of Theoretical Physics Vol. 83 No. 6 (1990) pp. 1090-1107
:
-
Fredholm Determinant Solution for the Inverse Scattering Transform of the N ×N Zakharov-Shabat Equation
-
Naruyoshi Asano and Yusuke Kato
