Fundamental Properties of the Binary Operators in Soliton Theory and Their Generalization

  • R. Hirota
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)

Abstract

We have introduced a binary operator to transform a class of nonlinear evolution equations into the bilinear form
$$F({D_x},{D_t},{\mkern 1mu} ...)f\cdot g = 0$$
The binary operator D x n operating on a pair of arbitrary functions f and g is defined by
$${D_x}^nf\left( x \right)\cdot g\left( x \right) = {\left( {\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial x'}}} \right)^n}f\left( x \right)g\left( {x'} \right)\left| {x'} \right. = x.$$
.

Keywords

Soliton Lution 

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References

  1. [1]
    Ryogo Hirota: “Direct Methods of Finding Exact Solutions of Nonlinear Evolution Equations”, in Bäcklund Transformations, ed. by R.M. Miura, Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, New York 1976) Vol. 515.Google Scholar
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    Ryogo Hirota and Junkichi Satsuma: Prog. Theor. Phys. Suppl. No. 59 (1976)64.Google Scholar
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    Ryogo Hirota: “Direct Methods in Soliton Theory”, in Solitons, ed. R.K. Bullough and P.J. Caudrey(Topics in Current Physics 17, Springer-Verlag, 1980 ).Google Scholar
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    Kazuhiro Nozaki and Nozaki Bekki:J.Phys. Soc. Jpn. 53 (1984) 1581.Google Scholar
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    Ryogo Hirota in preparation.Google Scholar
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    G.L. Lamb, Jr. and D.W. McLaughlin: “Aspects of Soliton Physics”, in Solitons, ed. R.K.Bullough and P.J. Caudrey (Topics in Current Physics 17, Springer-Verlag, 1980 ).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • R. Hirota
    • 1
  1. 1.Department of Applied Mathematics, Faculty of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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