Advertisement

Functional Analysis and Its Applications

, Volume 22, Issue 3, pp 200–213 | Cite as

Method of averaging for two-dimensional "integrable" equations

  • I. M. Krichever
Articles

Keywords

Functional Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    S. Yu. Dobrokhotov and V. P. Maslov, "Finite-zone almost-periodic solutions in WKB approximations," Current Problems in Mathematics (Summaries in Science and Technology) [in Russian], Vol. 15, Vsesoyuz. Inst. Nauchn. Tekh. Informatsii, Moscow (1980), pp. 3–94.Google Scholar
  2. 2.
    S. Yu. Dobrokhotov and V. P. Maslov, "Multiphase asymptotics of nonlinear partial differential equations with a small parameter," Soviet Scientific Reviews. Math. Phys. Reviews, Vol. 3, OPA, Amsterdam (1980), pp. 221–280.Google Scholar
  3. 3.
    H. Flaschka, M. Forest, and D. McLaughlin, "Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation," Commun. Pure Appl. Math.,33, No. 6, 739–784 (1980).Google Scholar
  4. 4.
    B. A. Dubrovin and S. P. Novikov, "Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov—Whitham averaging method," Dokl. Akad. Nauk SSSR,270, No. 4, 781–785 (1983).Google Scholar
  5. 5.
    S. P. Tsarev, "On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type," Dokl. Akad. Nauk SSSR,282, No. 3, 534–537 (1985).Google Scholar
  6. 6.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, "Nonlinear equations of Korteweg—de Vries type, finite-zone linear operators, and Abelian varieties," Usp. Mat. Nauk,31, No. 1, 55–136 (1976).Google Scholar
  7. 7.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, The Theory of Solitons: the Inverse Problem Method [in Russian], Nauka, Moscow (1980).Google Scholar
  8. 8.
    P. D. Lax, "Periodic solutions of the Korteweg—de Vries equation," Commun. Pure Appl. Math.,28, 141–188 (1975).Google Scholar
  9. 9.
    H. McKean and P. van Moerbeke, "The spectrum of Hill's equation," Invent. Math.,30, No. 3, 217–274 (1975).Google Scholar
  10. 10.
    V. E. Zakharov and A. B. Shabat, "A plan for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem method. 1," Funkts. Anal. Prilozhen.,8, No. 3, 43–53 (1974).Google Scholar
  11. 11.
    I. M. Krichever, "Algebraic-geometric construction of the Zakharov—Shabat equations and their periodic solutions," Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976).Google Scholar
  12. 12.
    I. M. Krichever, "Integration of nonlinear equations by the methods of algebraic geometry," Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).Google Scholar
  13. 13.
    B. A. Dubrovin, "The theta-function and nonlinear equations," Usp. Mat. Nauk,36, No. 2, 11–80 (1981).Google Scholar
  14. 14.
    I. M. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations," Usp. Mat. Nauk,32, No. 6, 183–208 (1977).Google Scholar
  15. 15.
    I. M. Krichever and S. P. Novikov, "Holomorphic bundles over algebraic curves, and nonlinear equations," Usp. Mat. Nauk,35, No. 6, 47–68 (1980).Google Scholar
  16. 16.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "Topological and algebraic geometry methods in contemporary mathematical physics. II," Soviet Scientific Reviews. Math. Phys. Reviews, Vol. 3, OPA, Amsterdam (1982), pp. 1–151.Google Scholar
  17. 17.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "Integrable systems. 1," Current Problems in Mathematics. Fundamental Directions (Summaries in Science and Technology) [in Russian], Vol. 4, Vsesoyuz. Inst. Nauchn. Tekh. Informatsii, Moscow (1985), pp. 179–285.Google Scholar
  18. 18.
    A. V. Gurevich and L. P. Pitaevskii, "Nonstationary structure of noncolliding shock waves," Zh. Éksp. Teor. Fiz.,65, 590–604 (1973).Google Scholar
  19. 19.
    I. M. Krichever, "A periodic solution of the Kadomtsev — Petviashvili equation," Dokl. Akad. Nauk SSSR,298, No. 4, 565–569 (1988).Google Scholar
  20. 20.
    E. I. Zverovich, "Boundary value problems in the theory of analytic functions," Usp. Mat. Nauk,26, No. 1, 113–181 (1971).Google Scholar
  21. 21.
    I. V. Cherednik, "Elliptic curves and matrix soliton differential equations," Algebra. Geometry. Topology (Summaries in Science and Technology) [in Russian], Vol. 22, Vsesoyuz. Inst. Nauchn. Tekh. Informatsii, Moscow (1984), pp. 205–265.Google Scholar
  22. 22.
    S. V. Manakov, "The method of the inverse problem of scattering theory and two-dimensional evolution equations," Usp. Mat. Nauk,31, No. 5, 245–246 (1976).Google Scholar
  23. 23.
    A. P. Veselov and S. P. Novikov, "Finite-zone, two-dimensional periodic Schrödinger operators. Explicit formulas and evolution equations," Dokl. Akad. Nauk SSSR,279, No. 1, 20–24 (1984).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • I. M. Krichever

There are no affiliations available

Personalised recommendations