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Integration of nonlinear equations by the methods of algebraic geometry

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Literature Cited

  1. C. Gardner, J. Greene, M. Kruskal, and R. Miura, "Method for solving the Korteweg—de Vries equation," Phys. Rev. Lett.,19, 1095–1098 (1967).

    Google Scholar 

  2. V. E. Zakharov and A. B. Shabat, "A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I," Funkts. Anal. Prilozhen.,8, 43–53 (1974).

    Google Scholar 

  3. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, "Nonlinear equations of the Korteweg—de Vries type, finite-zone linear operators, and Abelian manifolds," Usp. Mat. Nauk,31, No. 1, 55–136 (1976).

    Google Scholar 

  4. I. M. Krichever, "Algebrogeometric construction of the Zakharov—Shabat equations and their periodic solutions," Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976).

    Google Scholar 

  5. I. M. Krichever, "Algebraic curves and commutative matrix differential operators," Funkts. Anal. Prilozhen.,10, No. 2, 75–77 (1976).

    Google Scholar 

  6. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "The Schrödinger equation in a periodic field and Riemann surfaces," Dokl. Akad. Nauk SSSR,229, No. 1, 15–19 (1976).

    Google Scholar 

  7. S. P. Novikov, "The periodic problem for the Korteweg—de Vries equation. I," Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974).

    Google Scholar 

  8. I. M. Gel'fand and L. A. Dikii, "Asymptotic behavior of the resolvent of Sturm—Liouville operators and the algebra of Korteweg—de Vries equations," Usp. Mat. Nauk,30, No. 5, 67–101 (1975).

    Google Scholar 

  9. O. I. Bogoyavlenskii, "Integrals of higher stationary Korteweg—de Vries equations and eigenvalues of Hill's operator," Funkts. Anal. Prilozhen.,10, No. 2, 9–13 (1976).

    Google Scholar 

  10. I. R. Shafarevich, Foundations of Algebraic Geometry [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  11. I. M. Krichever, "Potentials with zero coefficient of reflection on a background of finite-zone potentials," Funkts. Anal. Prilozhen.,9, No. 2, 77–78 (1975).

    Google Scholar 

  12. G. Springer, Introduction to Riemann Surfaces, Addison-Wesley (1957).

  13. N. I. Akhiezer, "A continuous analog of orthogonal polynomials on a system of intervals," Dokl. Akad. Nauk SSSR,141, No. 2, 263–266 (1961).

    Google Scholar 

  14. B. A. Dubrovin, "Periodic problems for the Korteweg—de Vries equation in the class of finite-band potentials," Funkts. Anal. Prilozhen.,9, No. 3, 41–51 (1975).

    Google Scholar 

  15. A. R. Its and V. B. Matveev, "The Schrödinger operator in a finite-zone spectrum and N-soliton solutions of the Korteweg—de Vries equation," Teor. Mat. Fiz.,23, No. 1, 51–67 (1975).

    Google Scholar 

  16. B. A. Dubrovin and S. P. Novikov, "A periodicity problem for the Korteweg—de Vries and Sturm—Liouville equations. Their connection with algebraic geometry," Dokl. Akad. Nauk SSSR,219, No. 3, 19–22 (1974).

    Google Scholar 

  17. N. G. Chebotarev, Theory of Algebraic Functions [in Russian], Gostekhizdat, Moscow (1948).

    Google Scholar 

  18. E. I. Zverovich, "Boundary-value problems in the theory of analytic functions," Usp. Mat. Nauk,26, No. 1, 113–181 (1971).

    Google Scholar 

  19. A. R. Its and V. B. Matveev, "A class of solutions of the Korteweg—de Vries equation," in: Problems of Mathematical Physics [in Russian], No. 8, LGU (1976).

  20. B. B. Kadomtsev and V. I. Petviashvili, "Stability of combined waves in weakly dispersing media," Dokl. Akad. Nauk SSSR,192, No. 4, 753–756 (1970).

    Google Scholar 

  21. V. E. Zakharov, "On the problem of stochastization of one-dimensional chains of nonlinear oscillators," Zh. Eksp. Teor. Fiz.,65, No. 1, 219–225 (1973).

    Google Scholar 

  22. O. I. Bogoyavlenskii, On Perturbations of the Toda Lattice, Preprint, Chernogolovka (1976).

  23. P. D. Lax, "Periodic solutions of the KdV equations," Comm. Pure Appl. Math.,28, 141–188 (1975).

    Google Scholar 

  24. I. M. Gel'fand and L. A. Dikii, "Fractional powers of operators and Hamiltonian systems," Funkts. Anal. Prilozhen.,10, No. 4, 13–29 (1976).

    Google Scholar 

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G. M. Krzhizhanovskii Energy Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 11, No. 1, pp. 15–31, January–March, 1977.

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Krichever, I.M. Integration of nonlinear equations by the methods of algebraic geometry. Funct Anal Its Appl 11, 12–26 (1977). https://doi.org/10.1007/BF01135528

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  • DOI: https://doi.org/10.1007/BF01135528

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