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.
References:
| [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 · doi:10.1002/cpa.3160300106 |
| [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 · doi:10.1215/S0012-7094-89-05928-0 |
| [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 · doi:10.1007/BF01078578 |
| [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 · doi:10.1007/BF01001546 |
| [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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