Integrability pp 279-314

Part of the Lecture Notes in Physics book series (LNP, volume 767) | Cite as

Hirota’s Bilinear Method and Its Connection with Integrability

  • J. Hietarinta

Abstract

We give an introduction to Hirota’s bilinear method, which is particularly efficient for constructing multisoliton solutions to integrable nonlinear evolution equations. We discuss in detail how the method works for equations in the Korteweg–de Vries class and then go through some other classes of equations. Finally we discuss how the existence of multisoliton solutions can be used as an integrability condition and therefore as a method of searching for possible new integrable equations.

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References

  1. 1.
    R. Hirota, Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons, Phys. Rev. Lett. 27, 1192–1194, 1971.CrossRefADSMATHGoogle Scholar
  2. 2.
    R. Hirota, Exact Solution of the modified Korteweg-de Vries Equation for Multiple Collisions of Solitons, J. Phys. Soc. Japan 33, 1456–1459, 1972.ADSCrossRefGoogle Scholar
  3. 3.
    R. Hirota, Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons, J.Phys. Soc. Japan 33, 1459–1463, 1972.ADSCrossRefGoogle Scholar
  4. 4.
    R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14, 805–809, 1973.MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    R. Hirota, A New Form of Bäcklund Transformations and Its Relation to the Inverse Scattering Problem Progr, Theor. Phys. 52, 1498–1512, 1974.ADSMATHGoogle Scholar
  6. 6.
    M. Jimbo and T. Miwa, Solitons and Infinite Dimensional Lie Algebras Publ. RIMS, Kyoto Univ. 19, 943–1001, 1983.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Hirota, Direct Method of Finding Exact Solutions of Nonlinear Evolution Equations, in Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, M. Miura (ed.), Lecture Notes in Mathematics, vol 515, Springer, 40–68, 1976.Google Scholar
  8. 8.
    R. Hirota and J. Satsuma, A Variety of Nonlinear Network Equations Generated from the Bäcklund Transformation for the Toda Lattice. Progress Theoretical Phys., Suppl. 59 64–100, 1976.CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    R. Hirota, Direct methods in soliton theory. in Solitons, R.K. Bullough and P.J. Caudrey (eds.), Springer, 157–176, 1980.Google Scholar
  10. 10.
    J. Hietarinta, Hirota’s bilinear method and partial integrability, in Partially Integrable Equations in Physics, R. Conte and N. Boccara (eds.), Kluwer, 459–478, 1990.Google Scholar
  11. 11.
    B. Grammaticos, A. Ramani, and J. Hietarinta, Multilinear operators: the natural extension of Hirota’s bilinear formalism, Phys. Lett. A 190, 65–70, 1994.CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    J. Hietarinta, B. Grammaticos, and A. Ramani, Integrable Trilinear PDE’s, in Nonlinear Evolution Equations & Dynamical Systems, NEEDS ’94, V.G. Makhankov, A.R. Bishop and D.D. Holm (eds.), World Scientific, 54–63, 1995.Google Scholar
  13. 13.
    F. Kako and N. Yajima, Interaction of Ion-Acoustic Solitons in Two-Dimensional Space, J. Phys. Soc. Japan 49, 2063–2071, 1980Google Scholar
  14. 14.
    R. Hirota and M. Ito, Resonance of Solitons in One Dimension, J. Phys. Soc. Japan 52, 744–748, 1983Google Scholar
  15. 15.
    K. Ohkuma and M. Wadati, The Kadomtsev-Petviashvili Equation: the Trace Method and the Soliton Resonances, J. Phys. Soc. Japan 52, 749–760, 1983.CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    R. Hirota, Exact N-soliton solutions of wave equation of long waves in shallowwater in nonlinear lattices, J. Math. Phys. 14, 810–814, 1973.MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14, 805–809, 1973.MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    R. Radhakrishnan, M. Lakshmanan and J. Hietarinta, Inelastic collision and switching of coupled bright solitons in optical fibers, Phys. Rev. E 56, 2213–2216, 1997.CrossRefADSGoogle Scholar
  19. 19.
    R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A 85, 407–408, 1981Google Scholar
  20. 20.
    M. Boiti, J.J-P. Leon, L. Martina, and F. Pempinelli, Scattering of localized solitons on the plane, Phys. Lett. A 132, 432–439, 1988.CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    A.S. Fokas and P.M. Santini, Coherent Structures in Multidimensions, Phys. Rev. Lett. 63, 1329–1333, 1989Google Scholar
  22. 22.
    A.S. Fokas and P.M. Santini, Dromions and a boundary value problem for the Davey-Stewartson 1 equation, Physica D 44, 99–130, 1990.CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    J. Hietarinta and R. Hirota, Multidromion solutions to the Davey-Stewartson equation, Phys. Lett. A 145, 237–244, 1990.CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    C. Gilson and J.J.C. Nimmo, A direct method for dromion solution of the Davey- Stewartson equations and their asymptotic properties, Proc. R. Soc. Lond. A 435, 339–357, 1991.MATHADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Hietarinta and J. Ruokolainen, Dromions – The movie. (video animation, see http:// users.utu.fi/hietarin/dromions) 1990.
  26. 26.
    J. Hietarinta, One-dromion solutions for generic classes of equations, J, Phys. Lett. A 149, 113–118, 1990.CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    C. Gilson, Resonant behaviour in the Davey–Stewartson equation, Phys. Lett. A 161, 423–428, 1992.CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    R. Hirota and J. Satsuma, N-Soliton Solutions of Model Equations for Shallow Water Waves, J. Phys. Soc. Japan 40, 611–612, 1976Google Scholar
  29. 29.
    J. Satsuma and D.J. Kaup, A Bäcklund transformation for a Higher Order Korteweg-de Vries Equation, J. Phys. Soc. Japan 43, 692–697, 1977Google Scholar
  30. 30.
    R. Hirota, Reduction of soliton equations in bilinear form, Physica D 18, 161–170, 1986.MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    J. Hietarinta, Bäcklund transformations from the bilinear viewpoint, CRM Proceedings and Lecture Notes 29 245–251, 2001.MathSciNetGoogle Scholar
  32. 32.
    A. Newell and Z. Yunbo, The Hirota conditions, J. Math. Phys. 27, 2016–20121, 1986.MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28, 1732–1742, 1987.MATHCrossRefADSMathSciNetGoogle Scholar
  34. 34.
    M. Ito, An Extension of Nonlinear Evolution Equations of the K-dV (mK-dV) Type to Higher Orders, J. Phys. Soc. Jpn. 49, 771–778, 1980.CrossRefADSGoogle Scholar
  35. 35.
    A.C. Hearn, REDUCE User’s Manual Version 3.2, 1985.Google Scholar
  36. 36.
    B. Grammaticos, A. Ramani, and J. Hietarinta, A search for integrable bilinear equations: the Painlevé approach, J. Math. Phys. 31, 2572–2578, 1990.MATHCrossRefADSMathSciNetGoogle Scholar
  37. 37.
    J. Hietarinta, A search of bilinear equations passing Hirota’s three-soliton condition: II. mKdV-type bilinear equations, J. Math. Phys. 28, 2094–2101, 1987Google Scholar
  38. 38.
    J. Hietarinta, A search of bilinear equations passing Hirota’s three-soliton condition: III. sine-Gordon-type bilinear equations, J. Math. Phys. 28, 2586–2592, 1987.MATHCrossRefADSMathSciNetGoogle Scholar
  39. 39.
    J. Hietarinta, Recent results from the search for bilinear equations having threesoliton solutions, in Nonlinear evolution equations: integrability and spectral methods, A. Degasperis, A.P. Fordy and M. Lakshmanan (eds.), Manchester U.P., 307–317, 1990.Google Scholar
  40. 40.
    J. Hietarinta, A search of bilinear equations passing Hirota’s three-soliton condition: IV. Complex bilinear equations, J. Math. Phys. 29, 628–635, 1988.MATHCrossRefADSMathSciNetGoogle Scholar
  41. 41.
    R. Hirota and Y. Ohta, Hierarchies of Coupled Soliton Equations. I, J. Phys. Soc. Jpn. 60, 798–809, 1991.CrossRefADSMathSciNetMATHGoogle Scholar
  42. 42.
    J.J.C. Nimmo, Darboux transformations in two dimensions. in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, P. Clarkson (ed.), Kluwer Academic, 183–192, 1992.Google Scholar
  43. 43.
    V.K. Melnikov, A direct method for deriving a multisoliton solution for the problem of interaction of waves on the x, y plane, Commun. Math. Phys. 112, 639–652, 1987.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • J. Hietarinta
    • 1
  1. 1.Department of PhysicsUniversity of TurkuFinland

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