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Integrable pseudopotentials related to generalized hypergeometric functions. (English) Zbl 1274.37040
Summary: We construct integrable pseudopotentials with an arbitrary number of fields in terms of generalized hypergeometric functions. These pseudopotentials yield some integrable \((2+1)\)-dimensional hydrodynamic type systems. In two particular cases these systems are equivalent to integrable scalar 3-dimensional equations of second order. An interesting class of integrable \((1+1)\)-dimensional hydrodynamic type systems is also generated by our pseudopotentials.

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
17B63 Poisson algebras
17B80 Applications of Lie algebras and superalgebras to integrable systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] Zakharov V.E., Shabat A.B.: Integration of non-linear equations of mathematical physics by the inverse scattering method. Funct. Anal. Appl. 13(3), 13–22 (1979)
[2] Krichever I.M.: The \(\tau\)-function of the universal Whitham hierarchy, matrix models and topological field theories. Comm. Pure Appl. Math. 47(4), 437–475 (1994) · Zbl 0811.58064 · doi:10.1002/cpa.3160470403
[3] Dubrovin, B.A.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups. Lecture Notes in Mathematics, vol. 1620, pp. 120–348 (1996) · Zbl 0841.58065
[4] Odesskii A., Sokolov V.: On (2 + 1)-dimensional hydrodynamic-type systems possessing pseudopotential with movable singularities. Funct. Anal. Appl. 42(3), 205–212 (2008) · Zbl 1175.35028 · doi:10.1007/s10688-008-0029-z
[5] Odesskii A.V.: A family of (2 + 1)-dimensional hydrodynamic-type systems possessing pseudopotential. Selecta Math. (N.S.) 13(4), 727–742 (2008) · Zbl 1237.37048 · doi:10.1007/s00029-008-0050-3
[6] Pavlov M.V.: Classification of the Egorov hydrodynamic chains. Theor. Math. Phys. 138(1), 55–71 (2004)
[7] Odesskii A., Pavlov M.V., Sokolov V.V.: Classification of integrable Vlasov-type equations. Theor. Math. Phys. 154(2), 209–219 (2008) arXiv:0710.5655 · Zbl 1146.76061 · doi:10.1007/s11232-008-0020-0
[8] Bateman, H., Erdélyi, A.: Higher transcendental functions. Based, in part, on notes left by Harry Bateman, and compiled by the staff of the Bateman Manuscript Project. [Director: Erdélyi, A., Research associates: Magnus, W., Oberhettinger, F., Tricomi, F.G.] McGraw-Hill, New York (1953)
[9] Gelfand I.M., Graev M.I., Retakh V.S.: General hypergeometric systems of equations and series of hypergeometric type. Russian Math. Surv. 47(4), 1–88 (1992) · Zbl 0798.33010 · doi:10.1070/RM1992v047n04ABEH000915
[10] Ferapontov, E.V., Odesskii, A.V.: Integrable Lagrangians and modular forms. J. Geom. Phys. (to appear). arXiv:0707.3433 · Zbl 1188.35161
[11] Burovskiy, P.A., Ferapontov, E.V., Tsarev, S.P.: Second order quasilinear PDEs and conformal structures in projective space. arXiv:0802.2626 [nlin. SI] · Zbl 1198.35104
[12] Ames, W.F., Anderson, R.L., Dorodnitsyn, V.A., Ferapontov, E.V., Gazizov, R.K., Ibragimov, N.H., Svirshchevskii, S.R.: CRC handbook of Lie group analysis of differential equations, vol. 1. Symmetries, exact solutions and conservation laws. CRC Press, Boca Raton (1994)
[13] Ferapontov, E.V., Hadjikos, L., Khusnutdinova, K.R.: Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian. arXiv:0705.1774 · Zbl 1203.35215
[14] Gibbons J., Tsarev S.P.: Reductions of Benney’s equations, Phys. Lett. A 211, 19–24 (1996) · Zbl 1072.35588 · doi:10.1016/0375-9601(95)00954-X
[15] Gibbons J., Tsarev S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258, 263–270 (1999) · Zbl 0936.35184 · doi:10.1016/S0375-9601(99)00389-8
[16] Pavlov M.V.: Algebro-geometric approach in the theory of integrable hydrodynamic-type systems. Comm. Math. Phys. 272(2), 469–505 (2007) · Zbl 1131.37065 · doi:10.1007/s00220-007-0235-1
[17] Akhmetshin A.A., Krichever I.M., Volvovski Y.S.: A generating formula for solutions of associativity equations. Russian Math. Surv. 54(2), 427–429 (1999) · Zbl 0934.53032 · doi:10.1070/RM1999v054n02ABEH000135
[18] Ferapontov E.V., Khusnutdinova K.R.: On integrability of (2+1)-dimensional quasilinear systems. Comm. Math. Phys. 248, 187–206 (2004) · Zbl 1070.37047 · doi:10.1007/s00220-004-1079-6
[19] Ferapontov E.V., Khusnutdinova K.R.: The characterization of 2-component (2 + 1)-dimensional integrable systems of hydrodynamic type. J. Phys. A: Math. Gen. 37(8), 2949–2963 (2004) · Zbl 1040.35042 · doi:10.1088/0305-4470/37/8/007
[20] Ferapontov E.V., Marshal D.G.: Differential-geometric approach to the integrability of hydrodynamic chains: the Haanties tensor. Math. Ann. 339(1), 61–99 (2007) · Zbl 1145.37034 · doi:10.1007/s00208-007-0106-2
[21] Pavlov M.V.: Classification of integrable hydrodynamic chains and generating functions of conservation laws. J. Phys. A: Math. Gen. 39(34), 10803–10819 (2006) · Zbl 1098.35097 · doi:10.1088/0305-4470/39/34/014
[22] Ferapontov E.V., Khusnutdinova K.R., Tsarev S.P.: On a class of three-dimensional integrable Lagrangians. Comm. Math. Phys. 261(1), 225–243 (2006) · Zbl 1108.37045 · doi:10.1007/s00220-005-1415-5
[23] Tsarev S.P.: On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Soviet Math. Dokl., 31, 488–491 (1985) · Zbl 0605.35075
[24] Tsarev, S.P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izvestiya, 37(2), 397–419, 1048–1068 (1991)
[25] Pavlov M.V., Tsarev S.P.: Tri-Hamiltonian structures of the Egorov systems of hydrodynamic type. Funct. Anal. Appl. 37(1), 32–45 (2003) · Zbl 1019.37048 · doi:10.1023/A:1022971910438
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