Geodesic
Confluence of hypergeometric functions and integrable hydrodynamic-type systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 3, pp. 429-455 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct a new class of integrable hydrodynamic-type systems governing the dynamics of the critical points of confluent Lauricella-type functions defined on finite-dimensional Grassmannian $\mathrm{Gr}(2,n)$, i. e., on the set of $2\times n$ matrices of rank two. These confluent functions satisfy certain degenerate Euler–Poisson–Darboux equations. We show that in the general case, a hydrodynamic-type system associated with the confluent Lauricella function is an integrable and nondiagonalizable quasilinear system of a Jordan matrix form. We consider the cases of the Grassmannians $\mathrm{Gr}(2,5)$ for two-component systems and $\mathrm{Gr}(2,6)$ for three-component systems in detail.
Keywords: Lauricella function, integrable system.
Mots-clés : confluence 
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}
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Y. Kodama; B. G. Konopelchenko. Confluence of hypergeometric functions and integrable hydrodynamic-type systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 3, pp. 429-455. http://geodesic.mathdoc.fr/item/TMF_2016_188_3_a4/

[1] G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974 | MR | Zbl

[2] B. L. Rozhdestvenskii, N. N. Yanenko, Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[3] S. P. Tsarev, Izv. AN SSSR. Ser. matem., 54:5 (1990), 1048–1068 | DOI | MR | Zbl

[4] H. Flashka, M. G. Forest, D. W. McLaughlin, Commun. Pure Appl. Math., 33:6 (1980), 739–784 | DOI | MR

[5] V. E. Zakharov, Funkts. analiz i ego pril., 14:2 (1980), 15–24 | DOI | MR | Zbl

[6] Y. Kodama, B. G. Konopelchenko, W. K. Schief, J. Phys. A: Math. Theor., 48:22 (2015), 225202, 15 pp. | DOI | MR | Zbl

[7] M. V. Pavlov, J. Math. Phys., 44:9 (2003), 4134–4156 | DOI | MR | Zbl

[8] B. G. Konopelchenko, L. Martínez Alonso, E. Medina, J. Phys. A: Math. Theor., 43:43 (2010), 434020, 15 pp. | DOI | MR | Zbl

[9] B. A. Dubrovin, S. P. Novikov, UMN, 44:6(270) (1989), 29–98 | DOI | MR | Zbl

[10] M. V. Pavlov, TMF, 138:1 (2004), 55–70 | DOI | DOI | MR | Zbl

[11] A. V. Odesskii, V. V. Sokolov, Selecta Math. (N. S.), 16:1 (2010), 145–172 | DOI | MR | Zbl

[12] P. Appell, J. Math. Pures Appl. (3), 8 (1882), 173–216

[13] G. Lauricella, Rendiconti del Circolo Mat. Palermo, 7, Suppl. 1 (1893), 111–158 | DOI

[14] I. M. Gelfand, Dokl. AN SSSR, 288:1 (1986), 14–18 | MR | Zbl

[15] I. M. Gelfand, A. V. Zelevinskii, Funkts. analiz i ego pril., 20:3 (1986), 17–34 | DOI | MR | Zbl

[16] I. M. Gelfand, V. S. Retakh, V. V. Serganova, Dokl. AN SSSR, 298:1 (1988), 17–21 | MR | Zbl

[17] I. M. Gelfand, M. I. Graev, V. S. Retakh, UMN, 47:4(286) (1992), 3–82 | DOI | MR | Zbl

[18] P. Deligne, G. D. Mostow, Inst. Hautes Études Sci. Publ. Math., 63:1 (1986), 5–89 | DOI | MR | Zbl

[19] E. Looijenga, “Uniformization by Lauricella functions – an overview of the theory of Deligne–Mostow”, Arithmetic and Geometry Around Hypergeometric Functions, Progress in Mathematics, 260, eds. R.-P. Holzapfel, A. M. Uludaǧ, M. Yoshida, Birkhäuser, Basel, 2007, 207–244, arXiv: math/0507534 | DOI | MR | Zbl

[20] J. Stienstra, “GKZ hypergeometric structures”, Arithmetic and Geometry Around Hypergeometric Functions, Progress in Mathematics, 260, Birkhäuser, Basel, 2007, 313–371, arXiv: math/0511351 | DOI | MR | Zbl

[21] K. Aomoto, M. Kita, Theory of Hypergeometric Functions, Springer, Tokyo, 2011 | MR | Zbl

[22] H. Kimura, K. Takano, Tohoku Math. J. (2), 58:1 (2006), 1–31 | DOI | MR | Zbl

[23] Y. Kodama, “Integrability of hydrodynamic type equations”, Nonlinear World, Proceedings of the IV International Workshop on Nonlinear and Turbulent Processes in Physics (Kiev, USSR, October 9–22, 1989), v. 2, eds. A. G. Sitenko, V. E. Zakharov, V. M. Chernousenko, Naukova Dumka, Kiev, 1989, 115–118

[24] E. V. Ferapontov, Phys. D, 63:1–2 (1993), 50–70 | DOI | MR | Zbl

[25] E. V. Ferapontov, TMF, 99:2 (1994), 257–262 | DOI | MR | Zbl

[26] O. I. Mokhov, “Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations”, Topics in Topology and Mathematical Physics, American Mathematical Society Translations. Ser. 2, 170, ed. S. P. Novikov, AMS, Providence, RI, 1995, 121–151 | MR

[27] O. I. Mokhov, E. V. Ferapontov, Funkts. analiz i ego pril., 30:3 (1996), 62–72 | DOI | DOI | MR | Zbl

[28] Y. Kodama, B. G. Konopelchenko, J. Phys. A: Math. Gen., 35:31 (2002), L489–L500 | DOI | MR | Zbl