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

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


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
$${D_x}^nf\left( x \right)\cdot g\left( x \right) = {\left( {\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial x'}}} \right)^n}f\left( x \right)g\left( {x'} \right)\left| {x'} \right. = x.$$


Bilinear Form Binary Operator Nonlinear Evolution Equation Soliton Theory Linear Evolution Equation 
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  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
  2. [2]
    Ryogo Hirota and Junkichi Satsuma: Prog. Theor. Phys. Suppl. No. 59 (1976)64.Google Scholar
  3. [3]
    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
  4. [4]
    Kazuhiro Nozaki and Nozaki Bekki:J.Phys. Soc. Jpn. 53 (1984) 1581.Google Scholar
  5. [5]
    Allan P. Fordy and John Gibbons: Physics Letters 75A (1980) 325.CrossRefGoogle Scholar
  6. [6]
    Yuzo Hosono and Masayasu Mimura: J. Math. Kyoto Univ. 22–3 (1982) 435.Google Scholar
  7. [7]
    Ryogo Hirota in preparation.Google Scholar
  8. [8]
    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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