KdV ’95 pp 5-37 | Cite as

Integrability, Computation and Applications

  • M. J. Ablowitz
  • S. Chakravarty
  • B. M. Herbst


The study of integrable systems and the notion of integrability has been re-energized with the discovery that infinite-dimensional systems such as the Korteweg-de Vries equation are integrable. In this paper, the following novel aspects of integrability are described: (i) solutions of Darboux, Brioschi, Halphen-type systems and their relationships to monodromy problems and automorphic functions, (ii) computational chaos in integrable systems, (iii) we explain why we believe that homoclinic structures and homoclinic chaos associated with nonlinear integrable wave problems, will be observed in appropriate laboratory experiments.

Key words

integrability KdV equation nonlinear integrable wave functions monodromy problems automorphic functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ablowitz, M. J. and Segur, H.: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.zbMATHCrossRefGoogle Scholar
  2. 2.
    Novikov, S. P., Manakov, S. V., Pitaevskii, L. P., and Zakharov, V. E.: Theory of Solitons. The Inverse Scattering Method, Plenum, New York, 1984.zbMATHGoogle Scholar
  3. 3.
    Ablowitz, M. J. and Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.zbMATHCrossRefGoogle Scholar
  4. 4.
    Ablowitz, M. J., Chakravarty, S., and Takhtajan, L. A.: Comm. Math. Phys. 158 (1993), 289–314.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chakravarty, S., Ablowitz, M. J., and Clarkson, P. A.: sPhys. Rev. Lett. 65 (1990), 1085–1087.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Mason, L. J. and Woodhouse, N. M. J.: Nonlinearity 6 (1993), 569–581.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ablowitz, M. J., Chakravarty, S., and Takhtajan, L. A.: Integrable systems, self-dual Yang-Mills equations and connections with modular forms, in S. Xiao (ed.), Proc. of Nonlinear Problems in Engineering and Science, Science Press, Beijing, China, 1992.Google Scholar
  8. 8.
    Takhtajan, L. A.: Modular forms as τ-functions for certain integrable reductions of the Yang-Mills equations, in O. Babelon et al. (eds.), oc. Verdier Memorial Conference on Integrable Systems, Birkhauser, Berlin, 1993.Google Scholar
  9. 9.
    Chakravarty, S. and Ablowitz, M. J.: to be published, 1995.Google Scholar
  10. 10.
    Ercolani, N., Forest, M. G., and McLaughlin, D. W.: Physica D43 (1990), 349–384.MathSciNetGoogle Scholar
  11. 11.
    Ablowitz, M. J. and Herbst, B. M.: SIAM J. Appl. Math. 50 (1990), 339–251.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ablowitz, M. J., Schober, C. M., and Herbst, B. M.: Phys. Rev. Lett. 71 (1993), 2683–2686.CrossRefGoogle Scholar
  13. 13.
    Ablowitz, M. J., Herbst, B. M., and Schober, C. M.: On the numerical solution of the sine-Gordon equation, I. Integrable discretizations and homoclinic manifolds, PAM Report 214, University of Colorado, Boulder, 1994.Google Scholar
  14. 14.
    Ablowitz, M. J. and Schober, C. M.: Contemp. Math. 172 (1994), 253–268.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ward, R.S.: Phys. Lett. A 61 (1977), 81–82.MathSciNetCrossRefGoogle Scholar
  16. 15a.
    R. S. Ward and R. O. Wells: Twistor Geometry and Field Theory, Cambridge University Press, Cambridge, 1990.CrossRefGoogle Scholar
  17. 16.
    Chakravarty, S. and Ablowitz, M. J.: On reductions of self-dual Yang-Mills equations, in P. Winternitz and D. Levi (eds.), Proc. NATO Advanced Research Workshop, Plenum Press, New York, 1990.Google Scholar
  18. 17.
    Mason, D. and Sparling, G. A. J.: J. Geom. Phys. 8 (1991), 263–271.MathSciNetGoogle Scholar
  19. 18.
    Dubrovin, B.: Private communication.Google Scholar
  20. 19.
    Gibbons, G. W and Pope, C. N.: Commun. Math. Phys. 66 (1979), 267–290.MathSciNetCrossRefGoogle Scholar
  21. 20.
    Chazy, J.: C.R. Acad. Sci. Paris 149 (1909), 563–565.Google Scholar
  22. 21.
    Chakravarty, S.: To be published.Google Scholar
  23. 22.
    Takhtajan, L. A.: Commun. Math. Phys. 160 (1994), 295–315.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 23.
    Plemelj, J.: Problems in the Sense of Riemann and Klein, Interscience Publishers, New York, 1964.zbMATHGoogle Scholar
  25. 24.
    Birkhoff, G. D.: Collected Mathematical Papers, Vol. 1, Dover, New York, 1968.zbMATHGoogle Scholar
  26. 25.
    Jimbo, M, Miwa, T., and Ueno, K.: Physica D2 (1981), 306–352 and the references therein.MathSciNetGoogle Scholar
  27. 26.
    Jimbo, M. and Miwa, T.: Physica D2 (1981), 407–448.MathSciNetGoogle Scholar
  28. 27.
    Nehari, Z.: Conformal Mapping, McGraw-Hill, New York, 1952; (Reprinted by Dover, New York, 1975).zbMATHGoogle Scholar
  29. 28.
    Ablowitz, M. J. and Ladik, J. F: Stud. Appl Math. 55 (1976), 213–229.MathSciNetGoogle Scholar
  30. 29.
    Benjamin, T. B.: Proc. Roy. Soc. A 299 (1967), 59–75.CrossRefGoogle Scholar
  31. 30.
    Stokes, G. G.: Camb. Trans. 8 (1847), 441–473.Google Scholar
  32. 31.
    Whitham, G. B.: Linear and Nonlinear Waves, Wiley, New York, 1974.zbMATHGoogle Scholar
  33. 32.
    Zabusky, N. J. and Kruskal, M. D.: Phys. Rev. Lett. 15 (1965), 240–243.zbMATHCrossRefGoogle Scholar
  34. 33.
    Schober, C. M. and McLaughlin, D. W.: Physica D57 (1992), 447–465.MathSciNetGoogle Scholar
  35. 34.
    Lake, B. M., Yuen, H. C., Rungaldier, H., and Ferguson, W. E.: J. Fluid. Mech. 83 (1977), 49–74.CrossRefGoogle Scholar
  36. 35.
    Hasselmann, K.: Discussion, sroc. Royal Soc. London (A), 299 (1967), 67.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • M. J. Ablowitz
    • 1
  • S. Chakravarty
    • 1
  • B. M. Herbst
    • 2
  1. 1.Program in Applied MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of Applied MathematicsThe University of the Orange Free StateBloemfonteinSouth Africa

Personalised recommendations