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Elliptic solutions of the Korteweg-de Vries equation

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Literature cited

  1. 1.

    S. P. Novikov, “Periodic problem for the Korteweg-de Vries equation,” Funkts. Anal. Ego Prilozhen.,8, No. 3, 54–56 (1974).

  2. 2.

    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solutions: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).

  3. 3.

    B. A. Dubrovin and S. P. Novikov, “Periodic and conditionally periodic analogs of multisoliton solutions of the Korteweg-de Vries equation,” Zh. Éksp. Teor. Fiz.,67, No. 12, 2131–2143 (1974).

  4. 4.

    H. Airault, H. P. McKean, and J. Moser, “Rational and elliptic solutions of the KdV equation,” Commun. Pure Appl. Math.,30, 95–148 (1977).

  5. 5.

    A. R. Its and V. B. Matveev, “Hill operators with a finite number of lacunae and multisoliton solutions of the Korteweg-de Vries equation,” Teor. Mat. Fiz.,23, No. 1, 51–67 (1975).

  6. 6.

    N. N. Bogolyubov and Yu. A. Mitropol'skii, Asymptotic Methods in the Theory of Oscillations [in Russian], Nauka, Moscow (1974).

  7. 7.

    G. B. Whitham, “Nonlinear dispersive waves,” Proc. R. Soc.,139, 283–291 (1965).

  8. 8.

    H. Flashka, G. 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).

  9. 9.

    S. Yu. Dobrokhotov and V. P. Maslov, “Finite-zoned almost periodic solutions in WKB approximations,” in: Modern Problems of Mathematics, Vol. 15 [in Russian], VINITI, Moscow (1980), pp. 3–94.

  10. 10.

    B. A. Dubrovin and S. P. Novikov, “Algebraic-geometric Poisson brackets for real finitezoned solutions of the “sine-Gordon” equation and the nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR,267, No. 6, 1295–1300 (1982).

  11. 11.

    E. D. Belokolos and V. Z. Enol'skii, “Generalized ansatz of Lamb,” Teor Mat. Fiz.,53, No. 2, 271–282 (1982).

  12. 12.

    V. Z. Enol'skii, “Reduction of a two-zoned solution of the Korteweg-de Vries equation to elliptic functions by third order transformations,” Inst. Teor. Fiz. Preprint 83–112P, Kiev (1983).

  13. 13.

    E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Enol'skii, “Algebraic-geometric principles of superposition of finite-zoned solutions of integrable nonlinear equations,” Usp. Mat. Nauk,41, No. 2, 3–42 (1986).

  14. 14.

    M. V. Babich, A. I. Bobenko, and V. B. Matveev, “Solutions of nonlinear equations, integrable by the inverse problem method, in terms of Jacobi theta functions and symmetries of algebraic curves,” Izv. Akad. Nauk SSSR, Ser. Mat.,49, No. 3, 511–529 (1985).

  15. 15.

    M. V. Babich, A. I. Bobenko, and V. B. Matveev, “Reductions of multidimensional theta functions and symmetries of algebraic curves,” Dokl. Akad. Nauk SSSR,272, No. 1, 13–17 (1983).

  16. 16.

    V. B. Matveev and A. O. Smirnov, “On the simplest trigonal solutions of the Boussinesq and Kadomtsev-Petviashvili equations,” Dokl. Akad. Nauk SSSR,293, No. 1, 78–82 (1987).

  17. 17.

    A. O. Smirnov, “A matrix analog of Appell's theorem and reductions multidimensional Riemann theta functions,” Mat. Sb.,133, No. 3, 382–391 (1987).

  18. 18.

    V. B. Matveev and A. O. Smirnov, “On the Riemann theta function of a trigonal curve and solution of the Boussinesq and KP equations,” Lett. Math. Phys.,14, 25–31 (1987).

  19. 19.

    I. M. Krichever, “Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles,” Funkts. Anal. Ego Prilozhen.,14, No. 4, 45–54 (1980).

  20. 20.

    A. R. Its and V. Z. Enol'skii, “On the dynamics of the Calogero-Moser system and reductions of hyperelliptic integrals to elliptic integrals,” Funkts. Anal. Ego Prilozhen.,20, No. 1, 73–74 (1986).

  21. 21.

    E. D. Belokolos and V. Z. Enol'skii, “Verdier potentials and Weierstrass reduction theory,” Preprint of the Inst. of Metal Physics, Kiev (1988).

  22. 22.

    J.-L. Verdier, New Elliptic Solitons, Univ. of Wroclaw, Preprint No. 293, Wroclaw (1988).

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Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 66–73, June, 1989.

In conclusion the author wishes to thank A. R. Its, V. B. Matveev, I. M. Krichever, and V. Z. Enol'skii for their interest in his work and for useful discussions.

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Smirnov, A.O. Elliptic solutions of the Korteweg-de Vries equation. Mathematical Notes of the Academy of Sciences of the USSR 45, 476–481 (1989). https://doi.org/10.1007/BF01158237

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Keywords

  • Vries Equation
  • Elliptic Solution