Journal of Soviet Mathematics

, Volume 31, Issue 6, pp 3373–3387 | Cite as

Algebrotopological approach to the reality problem. Real action variables in the theory of finite-zone solutions of the sine-Gordon equation

  • S. P. Novikov
Article

Abstract

The paper develops an algebrotopological approach to the problem of effective selection of real finite-zone solutions of the sine-Gordon equation, which uses the socalledγ-representation on the Riemann surface, in which “action” variables can be computed explicitly. This approach is general and applies to many systems for which the reality problem has not yet been solved.

Keywords

Riemann Surface Reality Problem Action Variable Real Action Effective Selection 

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Literature cited

  1. 1.
    V. E. Zakharov and L. D. Fadeev, “The Korteweg-de Vries equation is a completely integrable Hamiltonian system,” Funkts. Anal. Prilozhen.,5, No. 4, 18–27 (1971).Google Scholar
  2. 2.
    S. P. Novikov (ed.), Soliton Theory. Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).Google Scholar
  3. 3.
    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
  4. 4.
    B. A. Dubrovin and S. M. Natanzon, “Real two-zone solutions of the sine-Gordon equation,” Funkts. Anal. Prilozhen.,16, No. 1, 27–43 (1982).Google Scholar
  5. 5.
    B. A. Dubrovin and S. P. Novikov, “Periodic and conditionally periodic analogs of multisoliton solutions of the Korteweg-de Vries (KdV) equation,” Zh. Eksp. Teor. Fiz.,67, No. 12, 2131–2143 (1974).Google Scholar
  6. 6.
    H. Flaschka and D. McLaughlin, “Canonically conjugate variables for KdV equation with periodic boundary conditions,” in: Progress Theor. Phys.,55, No. 2, 438–456 (1976).Google Scholar
  7. 7.
    A. P. Veselov and S. P. Novikov, “On Poisson brackets compatible with algebraic geometry and the dynamics of KdV on the set of finite-zone potentials,” Dokl. Akad. Nauk SSSR,266, No. 3, 233–237 (1982).Google Scholar
  8. 8.
    A. P. Veselov and S. P. Novikov, “Poisson brackets and complex tori,” Tr. Mat. Inst. AN SSSR,165, 39–62 (1983).Google Scholar
  9. 9.
    B. A. Dubrovin and S. P. Novikov, “Algebrogeometric Poisson brackets for real finitezone solutions of the nonlinear Schrödinger (NS) and sine-Gordon (SG) equations,” Dokl. Akad. Nauk SSSR,267, No. 6, 1295–1300 (1982).Google Scholar
  10. 10.
    S. I. Alber, “On stationary problems of Korteweg-de Vries type,” Commun. Pure Appl. Math.,34, 259–272 (1981).Google Scholar
  11. 11.
    V. A. Kozel and V. P. Kotlyarov, “Almost-periodic solutions of the equation utt-uxx+sin u=O,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 878–881 (1976).Google Scholar
  12. 12.
    A. R. Its and V. P. Kotlyarov, “Explicit formulas for the solutions of the nonlinear Schrödinger equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 965–968 (1976).Google Scholar
  13. 13.
    H. McKean, “The sine-Gordon and sinh-Gordon equations on the circle,” Commun. Pure. Appl. Math.,34, 197–257 (1981).Google Scholar
  14. 14.
    I. V. Cherednik, “On reality conditions in finite-zone integration,” Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).Google Scholar
  15. 15.
    N. Ercolani and M. G. Forest, “The geometry of real two-phase sine-Gordon wave trains,” Preprint Ohio State Univ., N 17/82 (1982).Google Scholar
  16. 16.
    B. A. Dubrovin, “Analytic properties of spectral data for non-self-adjoint operators, connected with real periodic solutions of the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,265, No. 4, 789–793 (1982).Google Scholar
  17. 17.
    I. M. Krichever, “Algebraic construction of Zakharov-Shabat equations and of their solutions,” Dokl. Akad. Nauk SSSR,277, No. 2, 291–294 (1976).Google Scholar
  18. 18.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Schrödinger equation in a periodic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR,229, No. 1, 15–18 (1976).Google Scholar
  19. 19.
    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
  20. 20.
    B. A. Dubrovin, “Matrix finite-zone operators,” in: Contemporary Problems in Mathematics [in Russian], Vol. 23, VINITI (1983), pp. 33–78.Google Scholar
  21. 21.
    S. P. Novikov, “The two-dimensional Schrödinger operator in periodic fields,” in: Contemporary Problems in Mathematics [in Russian], Vol. 23, VINITI (1983), pp. 3–32.Google Scholar
  22. 22.
    L. D. Faddeev and L. A. Takhtadzhyan, “Essentially nonlinear one-dimensional model of classical field theory,” Teor. Mat. Fiz.,21, No. 2, 160–174 (1974); supplement Teor. Mat. Fiz.,22, No. 1, 143–144 (1974).Google Scholar
  23. 23.
    B. A. Dubrovin and S. P. Novikov, “Hamiltonian formalism of one-dimensional systems of hydrodynamical type, and the Bogolyubov-Witham averaging method,” Dokl. Akad. Nauk SSSR,270, No. 4, 781–785 (1983).Google Scholar

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© Plenum Publishing Corporation 1985

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  • S. P. Novikov

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