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Theoretical and Mathematical Physics

, Volume 137, Issue 2, pp 1505–1514 | Cite as

The Infinite-Genus Limit of the Whitham Equations

  • G. A. El
Article
  • 34 Downloads

Abstract

We derive the infinite-genus limit of the KdV–Whitham equations based on the special scaling of the spectral curve introduced by Venakides in the study of the continuum limit of theta functions. The limit describes evolution of the integrated density of states in a one-dimensional soliton gas.

finite-gap potentials rotation number thermodynamic limit 

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REFERENCES

  1. 1.
    G. B. Whitham, Proc. Roy. Soc. A, 283, 238 (1965).Google Scholar
  2. 2.
    H. Flaschka, G. Forest, and D. W. McLaughlin, Comm. Pure Appl. Math., 33, 739 (1979).Google Scholar
  3. 3.
    P. D. Lax and C. D. Levermore, Comm. Pure Appl. Math., 36, 253, 571, 809 (1983).Google Scholar
  4. 4.
    S. Venakides and T. Am, Trans. Am. Math. Soc., 301, 189 (1987).Google Scholar
  5. 5.
    S. Venakides, Comm. Pure Appl. Math., 43, 335 (1990).Google Scholar
  6. 6.
    P. Deift, S. Venakides, and X. Zhou, Internat. Math. Res. Notices, No. 6, 285 (1997).Google Scholar
  7. 7.
    S. Venakides, Comm. Pure Appl. Math., 42, 711 (1989).Google Scholar
  8. 8.
    R. Johnson and J. Moser, Comm. Math. Phys., 84, 403 (1982).Google Scholar
  9. 9.
    G. A. El, A. L. Krylov, S. A. Molchanov, and S. Venakides, Phys. D, 152-153, 653 (2001).Google Scholar
  10. 10.
    V. E. Zakharov, JETP, 33, 538 (1971).Google Scholar
  11. 11.
    B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys, 44, 35 (1989).Google Scholar
  12. 12.
    S. P. Novikov, Funct. Anal. Appl., 8, 236 (1974).Google Scholar
  13. 13.
    P. D. Lax, Comm. Pure Appl. Math., 26, 141 (1975).Google Scholar
  14. 14.
    A. R. Its and V. B. Matveev, Theor. Math. Phys., 23, 343 (1975).Google Scholar
  15. 15.
    M. I. Weinstein and J. B. Keller, SIAM J. Appl. Math., 47, 941 (1987).Google Scholar
  16. 16.
    A. V. Gurevich, N. G. Mazur, and K. P. Zybin, JETP, 90, 695 (2000).Google Scholar
  17. 17.
    A. L. Krylov and G. A. El', Russ. Math. Surveys, 54, 439 (1999).Google Scholar
  18. 18.
    I. M. Krichever, Funct. Anal. Appl., 22, 200 (1988).Google Scholar
  19. 19.
    G. A. El, “Algebro-geometrical solutions of the infinite-genus Whitham equations,” to be published.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • G. A. El
    • 1
    • 2
  1. 1.School of Mathematical and Information SciencesCoventry UniversityCoventryUK
  2. 2.Ionosphere and Radio Wave Propagation, RAS, TroitskInstitute of Terrestrial MagnetismMoscow OblastRussia

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