Geodesic
The description of pairs of compatible first-order differential geometric poisson brackets
Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 2, pp. 293-309 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that bi-Hamiltonian structures of systems of hydrodynamic type can be described in terms of solutions of nonlinear equations. These equations can be integrated by the inverse scattering transform for arbitrary metrics as well as for flat ones. In particular, if one metric is flat and the other has a constant curvature, then the corresponding integrable system is a reduction of the Cherednik chiral-field model.
Keywords: Hamiltonian structure, system of hydrodynamic type.
Mots-clés : Riemann invariant 
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M. V. Pavlov. The description of pairs of compatible first-order differential geometric poisson brackets. Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 2, pp. 293-309. http://geodesic.mathdoc.fr/item/TMF_2005_142_2_a7/

[1] E. V. Ferapontov, “Gamiltonovy sistemy gidrodinamicheskogo tipa i ikh realizatsiya na giperpoverkhnostyakh psevdoevklidova prostranstva”, Itogi nauki i tekhniki. Problemy geometrii, 22, ed. N. M. Ostianu, VINITI, M., 1990, 59–96 | MR

[2] B. A. Dubrovin, S. P. Novikov, UMN, 44:6 (1989), 29–98 | MR | Zbl

[3] M. V. Pavlov, Dokl. RAN, 339:1 (1994), 21–23 ; 338:2, 165–167 | MR | Zbl | Zbl

[4] M. V. Pavlov, S. P. Tsarev, UMN, 46:4 (1991), 169–170 | MR

[5] O. I. Mokhov, UMN, 57:6 (2002), 189–190 ; Функц. анализ и его прилож., 36:3 (2002), 36–47 ; 96 ; УМН, 57:3(345) (2002), 155–156 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[6] E. V. Ferapontov, J. Phys. A, 34 (2001), 1–12 | DOI | MR

[7] B. A. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Groups, Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) (Montecatini Terme, Italy, June 14–22), Lecture Notes in Math., 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[8] A. Yu. Orlov, “Volterra operator algebra for zero curvature representation. Universality of KP”, Proceedings of the III Potsdam–Kiev Workshop at Clarkson University (Potsdam, NY, USA, August, 1–11, 1991), eds. A. S. Fokas, D. J. Kaup, A. C. Newell, V. E. Zakharov, Springer, Berlin, 1993, 126–131

[9] S. P. Tsarev, Dokl. AN SSSR, 282:3 (1985), 534–537 ; Изв. АН СССР. Сер. Матем., 54:5 (1991), 1048–1068 | MR | Zbl | MR

[10] E. V. Ferapontov, M. V. Pavlov, Physica D, 52 (1991), 211–219 | DOI | MR | Zbl

[11] G. Darboux, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, Gautier-Villar, Paris, 1910 | MR

[12] A. M. Kamchatnov, M. V. Pavlov, Phys. Lett. A, 301:3–4 (2002), 269–274 | DOI | MR | Zbl

[13] A. B. Borisov, S. A. Zykov, TMF, 115 (1998), 199–214 | DOI | MR | Zbl

[14] M. A. Kamchatnov, R. A. Kraenkel, J. Phys. A, 35 (2002), L13–L18 | DOI | MR | Zbl

[15] W. Oevel, W. Strampp, Commun. Math. Phys., 157 (1993), 51–81 | DOI | MR | Zbl

[16] E. V. Ferapontov, Funkts. analiz i ego prilozh., 25:3 (1991), 37–49 | MR | Zbl

[17] I. V. Cherednik, J. Nucl. Phys., 33 (1981), 278–282