Geodesic
Two-gap elliptic solutions of the Boussinesq equation
Sbornik. Mathematics, Tome 190 (1999) no. 5, pp. 763-781 Cet article a éte moissonné depuis la source Math-Net.Ru

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Two-gap solutions of the Boussinesq equation are considered. It is shown that for almost every Riemann surface $\Gamma$ of genus $g=2$ covering the elliptic surface it is possible to construct an elliptic (in $x$) two-gap solution of the Boussinesq equation. The existence of third- and fourth-order differential operators with elliptic “two-gap” potentials having an arbitrary number of poles is also established. An example is given. 
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A. O. Smirnov. Two-gap elliptic solutions of the Boussinesq equation. Sbornik. Mathematics, Tome 190 (1999) no. 5, pp. 763-781. http://geodesic.mathdoc.fr/item/SM_1999_190_5_a4/

[1] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1 (1976), 55–136 | MR | Zbl

[2] Matveev V. B., Abelian functions and solitons, Preprint No373, Univ. of Wrocław, 1976 | MR

[3] Krichever I. M., “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, UMN, 32:6 (1977), 183–208 | MR | Zbl

[4] Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov: metod obratnoi zadachi, Nauka, M., 1980 | MR

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

[6] Belokolos E. D., Bobenko A. I., Enol'skii V. Z., Its A. R., Matveev V. B., Algebro-geometrical approach to nonlinear evolution equations, Springer-Verlag, Berlin, 1994

[7] Airault H., McKean H. P., Moser J., “Rational and elliptic solutions of the KdV equation”, Comm. Pure Appl. Math., 30 (1977), 95–148 | DOI | MR | Zbl

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

[9] Eilbeck J. C., Enol'skii V. Z., “Elliptic Baker–Akhiezer functions and application to an integrable dynamical system”, J. Math. Phys., 35:3 (1994), 1192–1201 | DOI | MR | Zbl

[10] Perelomov A. M., Integriruemye sistemy klassicheskoi mekhaniki i algebry Li, Nauka, M., 1990 | Zbl

[11] Treibich A., Verdier J.-L., “Solitons elliptiques”, Progr. Math., 88 (1990), 437–480 | MR | Zbl

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

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

[14] Belokolos E. D., Enol'skii V. Z., “Reduction of theta function and elliptic finite-gap potentials”, Acta Appl. Math., 36:1–2 (1994), 87–117 | DOI | MR | Zbl

[15] Smirnov A. O., “Finite-gap elliptic solutions of the KdV equation”, Acta Appl. Math., 36 (1994), 125–166 | DOI | MR | Zbl

[16] Smirnov A. O., “Ellipticheskie resheniya nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya Kortevega–de Friza”, Matem. sb., 185:8 (1994), 103–114 | Zbl

[17] Smirnov A. O., “Dvukhzonnye ellipticheskie resheniya integriruemykh nelineinykh uravnenii”, Matem. zametki, 58:1 (1995), 86–97 | MR | Zbl

[18] Smirnov A. O., “Veschestvennye ellipticheskie resheniya uravnenii, svyazannykh s uravneniem sine-Gordon”, Algebra i analiz, 8:3 (1996), 196–211

[19] Gesztesy F., Weikard R., “Treibich–Verdier potentials and the stationary $(m)$KDV hierarchy”, Math. Z., 219:3 (1995), 451–476 | DOI | MR | Zbl

[20] Gesztesy F., Weikard R., A characterization of all elliptic solutions of the AKNS hierarchy, Preprint, 1997

[21] Gesztesy F., Holden H., A local sine-Gordon hierarchy and its algebro-geometric solutions, Preprint, 1997 | MR

[22] Smirnov A. O., “Ob odnom klasse ellipticheskikh reshenii uravneniya Bussineska”, TMF, 109:3 (1996), 347–356 | MR | Zbl

[23] Brezhnev Yu. V., “Darboux transformation and some multi-phase solutions of the Dodd–Bullough–Tzitzeica equation: $U_{xt}=e^U-e^{-2U}$”, Phys. Lett. A, 211 (1996), 94–100 | DOI | MR | Zbl

[24] Zverovich E. I., “Kraevye zadachi teorii analiticheskikh funktsii v gelderovskikh klassakh na rimanovykh poverkhnostyakh”, UMN, 26:1 (1971), 113–179 | MR | Zbl

[25] Igusa J., Theta functions, Die Grundlehren der mathematischen Wissenschaften, 194, Springer-Verlag, Berlin, 1972 | MR | Zbl

[26] Fay J. D., Theta-functions on Riemann surfaces, Lect. Notes in Math., 352, 1973 | MR | Zbl

[27] Matveev V. B., Smirnov A. O., “O prosteishikh trigonalnykh resheniyakh uravnenii Bussineska i Kadomtseva–Petviashvili”, Dokl. AN SSSR, 293:1 (1987), 78–82 | MR | Zbl

[28] Springer Dzh., Vvedenie v teoriyu rimanovykh povekhnostei, IL, M., 1960

[29] Smirnov A. O., “Ellipticheskie po $t$ resheniya uravneniya KdF”, TMF, 100:2 (1994), 183–198 | MR | Zbl

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

[31] Kurant A., Gurvits R., Teoriya funktsii, Nauka, M., 1968 | MR