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The elliptic-in-\(t\) solutions of the nonlinear Schrödinger equation. (English. Russian original) Zbl 0927.35110

Theor. Math. Phys. 107, No. 2, 568-578 (1996); translation from Teor. Mat. Fiz. 107, No. 2, 188-200 (1996).
Summary: Four various “ansatzes” of the Krichever curves for the periodic-in-\(t\) solutions of the nonlinear Schrödinger equation are considered. An example is given.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
14H52 Elliptic curves
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[1] S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov,Theory of Solitons. The Method of the Inverse Scattering, Plenum, New York (1984).
[2] L. D. Faddeev and L. A. Takhtajan,The Hamiltonian Methods in Theory of Solitons, Berlin-Heidelberg-New York, Springer (1987). · Zbl 1111.37001
[3] F. Calogero and A. Degasperis,Solitons and Spectral Transform I, North-Holland, Amsterdam (1982). · Zbl 0501.35072
[4] M. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1985). · Zbl 0472.35002
[5] R. Dodd, J. Eilbuck, J. Gibbon, and H. Morris,Solitons and Nonlinear Wave Equations [Russian translation], Mir, Moscow (1988).
[6] A. C. Newell,Solitons in Mathematics and Physics, SIAM, Philadelphia (1985). · Zbl 0565.35003
[7] B. A. Dubrovin and S. P. Novikov,Zh. Exp. Theor. Phys.,67, 2131 (1974).
[8] H. Airault, H. P. McKean, and J. Moser,Commun. Pure Appl. Math.,30, 95 (1977). · Zbl 0338.35024
[9] I. M. Krichever,Funkts. Anal. Prilozh.,14, No. 4, 45 (1980).
[10] E. D. Belokolos and V. Z. Enolskii,Teor. Mat. Fiz.,53, 271 (1982).
[11] M. V. Babich, A. I. Babenko, and V. B. Matveev,Izv. Akad. Nauk SSSR, ser. Mat.,49, 511 (1985).
[12] E. D. Belokolos, A. I. Babenko, V. B. Matveev, and V. Z. Enolskii,Usp. Mat. Nauk,41, No. 2, 3 (1986).
[13] J.-L. Verdier, in:Algebraic Analysis, Vol. II, Academic Press (1988), pp. 901.
[14] A. Treibich,Duke Math. J.,59, 611 (1989). · Zbl 0698.14029
[15] A. Treibich and J.-L. Verdier,Solitons elliptiques (Volume en l’honneur du anniversaire du prof. A. Grothendieck), Birkhäuser, Boston (1990).
[16] E. D. Belokolos and V. Z. Enolskii,Usp. Mat. Nauk,44, No. 5, 155 (1989).
[17] E. D. Belokolos and V. Z. Enolskii,Funkts. Anal. Prilozhen.,23, No. 1, 57 (1989). · Zbl 0701.22008
[18] A. O. Smirnov,Mat. Zametki,46, No. 5, 100 (1989).
[19] A. O. Smirnov,Mat. Zametki,45, No. 6, 66 (1989).
[20] I. A. Taimanov,Teor. Mat. Fiz.,84, 38 (1990).
[21] A. O. Smirnov,Teor. Mat. Fiz.,100, 183 (1994).
[22] A. O. Smirnov,Acta Appl. Math.,36, 125 (1994). · Zbl 0828.35119
[23] I. M. Krichever,Usp. Mat. Nauk,33, No. 4, 215 (1978).
[24] I. M. Krichever,Usp. Mat. Nauk,36, No. 2, 79 (1981).
[25] A. O. Smirnov,Teor. Mat. Fiz.,78, 11 (1989).
[26] A. O. Smirnov,Mat. Sbornik,185, No. 8, 103 (1994).
[27] A. I. Babenko,Funkts. Anal. Prilozhen.,18, No. 3, 74 (1984).
[28] A. O. Smirnov,Mat. Sbornik,181, No. 6, 804 (1990).
[29] A. O. Smirnov,Zap. Nauchn. Sem. POMI,205 (1993).
[30] A. R. Its,Vestnik Leningr. Gos. Univ., Mat. Mekh. Astr.,7, No. 2, 39 (1976).
[31] A. R. Its, ”The exact integration in the Riemann {\(\theta\)}-functions of the nonlinear Schrödinger equation and of the modified Korteweg-de Vries equation,” Doctoral thesis, Leningr. Gos. Univ., Leningrad (1977).
[32] A. R. Its and V. P. Kotliarov,Dokl. Akad. Nauk UkrSSR, ser. A,11, 965 (1976).
[33] V. B. Matveev, ”Abelian functions and solitons,” Preprint No. 373, Wroclaw, Univ. of Wroclaw (1976).
[34] A. R. Its,Izv. Acad. Nauk USSR, Ser. Mat.,49, 530 (1985).
[35] A. Krazer,Lehrbuch der Thetafunktionen, Teubner, Leipzig (1903).
[36] N. I. Akhiezer,Elements of Elliptic Function Theory [in Russian], Nauka, Moscow (1970). Translated by V. I. Serdobol’skii.
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