Prog. Theor. Phys. Vol. 78 No. 2 (1987) pp. 198-213
Fredholm Determinants and the Cauchy Problem of a Class of Nonlinear Evolution Equations
Department of Engineering Mathematics, Utsunomiya University, Utsunomiya 321
(Received January 19, 1987)
The Marchenko equation for the Zakharov-Shabat operator is studied in the framework of the Fredholm theory and the solution, the Zakharov-Shabat potential, is presented by the Fredholm determinant and the first minors, which are composed of the kernel of the Marchenko equation. The evolution of the potential due to the evolution of the kernel is proved to produce the solution of the Cauchy problem for a class of nonlinear evolution eqations including the nonlinear Schrödinger equation and the modified Korteweg-de Vries equation. In Hirota's formulation by bilinear forms we separate the general identity associated to the redundancy in the inverse scattering method for the Zakharov-Shabat operator and show that the formulation is valid also in cases containing ripples in addition to solitons.
DOI : 10.1143/PTP.78.198
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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