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Prog. Theor. Phys. Vol. 78 No. 2 (1987) pp. 198-213

[ Full Text PDF : FREE ACCESS (568K) ]

Fredholm Determinants and the Cauchy Problem of a Class of Nonlinear Evolution Equations

Yusuke Kato

Department of Engineering Mathematics, Utsunomiya University, Utsunomiya 321

(Received January 19, 1987)

Abstract:

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.


URL : http://ptp.ipap.jp/link?PTP/78/198/
DOI : 10.1143/PTP.78.198

[ Full Text PDF : FREE ACCESS (568K) ] Citation:

References:

  1. Y. Kato, Prog. Theor. Phys. 76 (1986), 768[PTP].
  2. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Stud. App. Maths. 53 (1974), 249.
  3. Y. Kato and H. Nakajima, Prog. Theor. Phys. 76 (1986), 981[PTP]; and also UTUP-36, Utsunomiya University, Nov. 1985 (unpublished).
  4. R. Hirota, Phys. Rev. Lett. 27 (1971), 1192[APS].
  5. R. Hirota, J. Phys. Soc. Jpn. 33 (1972), 1456[JPSJ].
  6. R. Hirota, J. Math. Phys. 14 (1973), 805[CrossRef].
  7. N. Asano and Y. Kato, Prog. Theor. Phys. 58 (1977), 61[PTP].
  8. N. Asano and Y. Kato, Prog. Theor. Phys. 74 (1985), 1005[PTP].
  9. N. Asano and Y. Kato, in Proceedings of International Conference on Plasma Physics (Fusion Research Assoc. of Japan, Nagoya, 1980), Vol. I, p. 87.
  10. Y. Kato, Inverse Problem in the Scattering Theory (in Japanese, Iwanami, Tokyo 1983).
  11. N. Asano and Y. Kato, J. Math. Phys. 25 (1984), 570[CrossRef].

Citing Article(s) :

  1. 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

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