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
Laboratory of Applied Mathematics and Mathematical Physics
Department of Information Science, Faculty of Engineering, Utsunomiya
University, Utsunomiya 321
(Received May 6, 1988)
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
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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