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2007 | 112 | 6 | 1171-1184
Article title

Hints on the Hirota Bilinear Method

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EN
Abstracts
EN
We discuss four stages of the Hirota bilinear method, for construction of soliton solutions to partial differential equations: the proper substitution to express the equation in the bilinear variables (1), reduction of the excess degrees of freedom (2), the perturbation scheme (3), and solution of the system of equations at the successive orders of magnitude (4). For the first stage we suggest an extension of the well-known singularity analysis. In the second stage we point out the need for caution to avoid overdetermined systems. In the third one we suggest a path to proper assumptions on the orders of magnitude of the unknown functions. Finally, we summarize the question of the choice of appropriate solutions. For the expansions at the stages (1) and (3) we suggest a 'renormalization', i.e. completion of the lower order terms with higher order ones to achieve the desired form of the coefficients.
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Year
Volume
112
Issue
6
Pages
1171-1184
Physical description
Dates
published
2007-12
received
2007-10-11
Contributors
author
  • The Andrzej Soltan Institute for Nuclear Studies, HoŻa 69, 00-681 Warsaw, Poland
References
  • 1. R. Hirota, in: Solitons , Eds. R.K. Bullough, P.J. Caudrey, Springer, Berlin 1980, p. 157
  • 2. J.S. Russell, Report on Waves, Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844, John Murray, London 1845, p. 311, Plates XLVII-LVII
  • 3. D.J. Korteweg, G. de Vries,. Philos. Mag. 39 , 422 (1895)
  • 4. N. Zabusky, M.D. Kruskal, Phys. Rev. Lett., 15, 240 (1965)
  • 5. P.D. Lax, Commun. Pure Appl. Math., 21, 467 (1968); the result was mentioned earlier in [6] as private communication
  • 6. C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett., 19, 1095 (1967)
  • 7. I. Kay, H.E. Moses, J. Appl. Phys., 27, 1503 (1956)
  • 8. V.E. Zakharov, A.B. Shabat, Zh. Eksp. Teor. Fiz., 61, 118 (1971) [Sov. Phys. JETP, 34, 62 (1972)]
  • 9. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett., 31, 125 (1973); Stud. Appl. Math., 53, 249 (1974)
  • 10. R. Hirota, Phys. Rev. Lett., 27, 1192 (1971)
  • 11. R. Hirota, J. Phys. Soc. Jpn., 33, 1456 (1972)
  • 12. R. Hirota, J. Phys. Soc. Jpn., 33, 1459 (1972)
  • 13. R. Hirota, J. Math. Phys., 14, 805 (1972)
  • 14. G.A. Baker Jr., P. Graves-Morris, Pade Approximants, Part I, Basic Theory (Encyclopedia of Mathematics and its Applications, Vol. 13), Addison-Wesley, Reading Mass. 1981, ch. 1
  • 15. B. Grammaticos, A. Ramani, J. Hietarinta, Phys. Lett. A, 190, 65 (1994)
  • 16. P.G. Estevez, P.R. Gordos, L. Martinez Alonso, E. Medina Reus, J. Phys A, Math. Gen., 26, 1915 (1993)
  • 17. J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys., 24, 522 (1983)
  • 18. J. Weiss, J. Math. Phys., 25, 2226 (1984)
  • 19. R. Conte, in: The Painleve Property One Century Later , Ed. R. Conte, Springer, New York 1999, p. 77
  • 20. K. Porsezian, K. Nakkeran, Phys. Rev. Lett., 76, 3995 (1996)
  • 21. M. Gedalin, T.C. Scott, Y.B. Band, Phys. Rev. Lett., 78, 448 (1997)
  • 22. P.P. Goldstein, Acta Phys. Pol. A, 110, 769 (2006)
  • 23. V. Cao Long, P.P. Goldstein, M. Trippenbach, Acta Phys. Pol. A, 105, 437 (2004)
  • 24. V. Cao Long, P.P. Goldstein, S. Vu Ngoc, Acta Phys. Pol. A, 106, 843 (2004)
  • 25. J. Hietarinta, J. Math. Phys., 28, 1732, 2094, 2586 (1988); 29, 1732, (1989)
  • 26. P. Goldstein, E. Infeld, Phys. Lett. A, 103, 8 (1984)
  • 27. P.P. Goldstein, Phys. Lett. A, 121, 11 (1987)
  • 28. H.C. Morris, J. Mat. Phys., 28, 530 (1977)
  • 29. S.-J. Yu, K. Toda, N. Sasa, T. Fukuyama, J. Phys. A, Math. Gen., 31, 3337 (1998)
  • 30. H.H. Dai, X. Geng, J. Math. Phys., 41, 7501 (2000); H.H. Dai, E.G. Fan, X.G. Geng, arXiv:nlin/0602015 (2006)
Document Type
Publication order reference
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YADDA identifier
bwmeta1.element.bwnjournal-article-appv112n602kz
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