KdV ’95 pp 361-378 | Cite as

Symbolic Software for Soliton Theory

  • W. Hereman
  • W. Zhuang


Four symbolic programs, in Macsyma or Mathematica language, are presented. The first program tests forthe existence of solitons for nonlinear PDEs. It explicitly constructs solitons using Hirota’s bilinear method. In the second program, the Painlevé integrability test for ODEs and PDEs is implemented. The third program provides an algorithm to compute conserved densities for nonlinear evolution equations. The fourth software package aids in the computation of Lie symmetries of systems of differential and difference-differential equations. Several examples illustrate the capabilities of the software.

Key words

soliton theory symbolic programs Hirota method Painlevé test Lie symmetries conserved densities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ablowitz, M. J. and Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics, Vol. 149, Cambridge University Press, Cambridge, UK, 1991.Google Scholar
  2. 2.
    Ablowitz, M. J. and Segur, H.: Solitons and the Inverse Scattering, SIAM Studies in Appl Math. Vol. 4, SIAM, Philadelphia, 1981.Google Scholar
  3. 3.
    Champagne, B., Hereman, W., and Winternitz, P.: The computer calculation of Lie point symmetries of large systems of differential equations, Comp. Phys. Comm. 66 (1991), 319–340.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Drazin, P. G. and Johnson, R. S.: Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.zbMATHCrossRefGoogle Scholar
  5. 5.
    Hereman, W.: SYMMGRPMAX and other symbolic programs for symmetry analysis of partial differential equations, in: E. Allgower, K. Georg and R. Miranda (Eds), Exploiting Symmetry in Applied and Numerical Analysis, Lectures in Applied Mathematics, Vol. 29, American Mathematical Society, Providence, Rhode Island, pp. 241–257, 1993.Google Scholar
  6. 6.
    Hereman, W.: Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath Bulletin 1(2) (1994), 45–82.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hereman, W.: Symbolic software for Lie symmetry analysis, in: N. H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, CRC Press, Boca Raton, Florida, Ch. 13, in press (1995).Google Scholar
  8. 8.
    Hereman, W. and Angenent, S.: The Painlevé test for nonlinear ordinary and partial differential equations, MACSYMA Newsletter 6 (1989), 11–18.Google Scholar
  9. 9.
    Hereman, W. and Zhuang, W.: Symbolic computation of solitons with MACSYMA, in: W. F. Ames and P. J. van der Houwen (Eds), Computational and Applied Mathematics II: Differential Equations, North-Holland, Amsterdam, The Netherlands, pp. 287–296, 1992.Google Scholar
  10. 10.
    Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition, Part I. KdV-type bilinear equations, J. Math. Phys. 28 (1987), 1732–1742.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hirota, R.: Direct method of finding exact solutions of nonlinear evolution equation, in: R. M. Miura (Ed.), Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, Lecture Notes in Mathematics, Vol. 515, Springer-Verlag, Berlin, pp. 40–68, 1976.CrossRefGoogle Scholar
  12. 12.
    Hirota, R.: Direct methods in soliton theory, in: R. K. Bullough and P. J. Caudrey (Eds), Solitons, Topics in Physics, Vol. 17, Springer-Verlag, Berlin, pp. 157–176, 1980.CrossRefGoogle Scholar
  13. 13.
    Kruskal, M. D., Miura, R. M., Gardner, C. S., and Zabusky, N. J.: Korteweg-de Vries equations and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys. 11 (1970), 952–960.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Matsuno, Y.: Bilinear Transformation Method, Academic Press, Orlando, 1984.zbMATHGoogle Scholar
  15. 15.
    Miura, R. M., Gardner, C. S., and Kruskal, M. D.: Korteweg-de Vries equations and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968), 1204–1209.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Murray, J. D.: Mathematical Biology, Biomathematics Texts, Vol. 19, Springer-Verlag, Berlin, 1989.Google Scholar
  17. 17.
    Newell, A. C: Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985.CrossRefGoogle Scholar
  18. 18.
    Newell, A. C. and Zeng, Y. B.: The Hirota conditions, J. Math. Phys. 29 (1987), 2016–2021.Google Scholar
  19. 19.
    Olver, P. J.: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York (1986 and 1993 2nd edition).CrossRefGoogle Scholar
  20. 20.
    Steeb, W.-H. and Euler, N.: Nonlinear Evolution Equations and Painlevé Test, World Scientific, Singapore, 1988.zbMATHGoogle Scholar
  21. 21.
    Verheest, F. and Hereman, W.: Conservation laws and solitary wave solutions for generalized Schamel equations, Physica Scripta 50 (1994), 611–614.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Willox, R., Hereman, W., and Verheest, F.: Complete integrability of a modified vector derivative nonlinear Schrödinger equation, Physica Scripta 51 (1995) in press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • W. Hereman
    • 1
  • W. Zhuang
    • 1
  1. 1.Department of Mathematical and Computer SciencesColorado School of MinesGoldenUSA

Personalised recommendations