Geodesic
The Dirac operator with elliptic potential
Sbornik. Mathematics, Tome 186 (1995) no. 8, pp. 1213-1221 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Dirac operator with elliptic finite-gap potential $$ -\mathrm i\begin{pmatrix}1&0\\0&-1\end{pmatrix}\Psi _x +\begin{pmatrix}0&p\\q&0\end{pmatrix}\Psi =\lambda\Psi . $$ is considered. An Ansatz for the Krichever curves associated with elliptic (in $x$) finite-gap solutions of the 'decomposed' non-linear Schrödinger equation $$ \begin{cases} \mathrm ip_t+p_{xx}-2p^2q=0, \\iq_t-q_{xx}+2pq^2=0 \end{cases} $$ and of the modified $KdV$ ($mKdV$) equation $$ \begin{cases} p_t+p_{xxx}-6pqp_x=0, \\q_t+q_{xxx}-6pqq_x=0. \end{cases} $$ is presented. Examples of two- and three-sheeted coverings associated with the one- and twogap Dirac potential are discussed. 
@article{SM_1995_186_8_a6,
     author = {A. O. Smirnov},
     title = {The {Dirac} operator with elliptic potential},
     journal = {Sbornik. Mathematics},
     pages = {1213--1221},
     year = {1995},
     volume = {186},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/}
}
TY  - JOUR
AU  - A. O. Smirnov
TI  - The Dirac operator with elliptic potential
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 1213
EP  - 1221
VL  - 186
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/
LA  - en
ID  - SM_1995_186_8_a6
ER  - 
%0 Journal Article
%A A. O. Smirnov
%T The Dirac operator with elliptic potential
%J Sbornik. Mathematics
%D 1995
%P 1213-1221
%V 186
%N 8
%U http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/
%G en
%F SM_1995_186_8_a6
A. O. Smirnov. The Dirac operator with elliptic potential. Sbornik. Mathematics, Tome 186 (1995) no. 8, pp. 1213-1221. http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/

[1] Krichever I. M., “Ellipticheskie resheniya uravneniya Kadomtseva–Petviashvili i integriruemye sistemy chastits”, Funktsion. analiz i ego pril., 14:4 (1980), 45–54 | MR | Zbl

[2] Belokolos E. D., Bobenko A. I., Matveev V. B., Enolskii V. Z., “Algebrogeometricheskie printsipy superpozitsii konechnozonnykh reshenii integriruemykh nelineinykh uravnenii”, UMN, 41:2 (1986), 3–42 | MR | Zbl

[3] Belokolos E. D., Enolskii V. Z., “Izospektralnye deformatsii ellipticheskikh potentsialov”, UMN, 44:5 (1989), 155–156 | MR | Zbl

[4] Belokolos E. D., Enolskii V. Z., “Ellipticheskie solitony Verde i teoriya reduktsii Veiershtrassa”, Funktsion. analiz i ego pril., 23:1 (1989), 57–58 | MR | Zbl

[5] Smirnov A. O., “Ellipticheskie resheniya integriruemykh nelineinykh uravnenii”, Matem. zametki, 46:5 (1989), 100–102 | MR | Zbl

[6] Smirnov A. O., “Ellipticheskie resheniya uravneniya Kortevega–de Friza”, Matem. zametki, 45:6 (1989), 66–73 | MR | Zbl

[7] Verdier J.-L., “New elliptic solitons”, Algebraic Analisys, V. II, Micio Sato Sixtieth Birthday Vol., Academic Press, 1988, 901–910 | MR

[8] Treibich A., “Tangential polinomials and elliptic solitons”, Duke Math. J., 59:3 (1989), 611–627 | DOI | MR | Zbl

[9] Treibich A., Verdier J.-L., Solitons elliptiques, Volume en l'honneur du $60^{\text e}$ anniversaire du prof. A. Grothendieck, Birkhäuser, Boston, 1990 | MR | Zbl

[10] Smirnov A. O., “Veschestvennye ellipticheskie resheniya uravneniya “sine-Gordon””, Matem. sb., 181:6 (1990), 804–812 | Zbl

[11] Smirnov A. O., 3-ellipticheskie resheniya uravneniya “sine-Gordon”, Matem. zametki, 62, no. 3, 1997 | MR | Zbl

[12] Its A. R., “Obraschenie giperellipticheskikh integralov i integrirovanie nelineinykh differentsialnykh uravnenii”, Vestnik LGU. Ser. Matem.-mekh.-astr., 7:2 (1976), 39–46 | MR | Zbl

[13] Its A. R., Tochnoe integrirovanie v rimanovykh $\Theta $-funktsiyakh nelineinogo uravneniya Shrëdingera i modifitsirovannogo uravneniya Kortevega–de Friza, Dis. ...kand. fiz.-matem. nauk, LGU, L., 1977

[14] Its A. R., Kotlyarov V. P., “Ob odnom klasse reshenii nelineinogo uravneniya Shrëdingera”, DAN USSR. Ser. A, 1976, no. 11, 965–968 | MR | Zbl

[15] Matveev V. B., Abelian functions and solitons, Preprint No 373, Univ. of Wrocław, 1976

[16] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970 | MR | Zbl