Geodesic
Hypergeometric Solutions of Soliton Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 128 (2001) no. 1, pp. 84-108
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We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations. 
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A. Yu. Orlov; D. M. Shcherbin. Hypergeometric Solutions of Soliton Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 128 (2001) no. 1, pp. 84-108. http://geodesic.mathdoc.fr/item/TMF_2001_128_1_a7/

[1] V. S. Dryuma, Pisma v ZhETF, 19 (1974), 653–754

[2] V. E. Zakharov, A. B. Shabat, Funkts. analiz i ego prilozh., 8:3 (1974), 43–53 ; 13:3 (1979), 13–22 | MR | Zbl | MR | Zbl

[3] A. V. Mikhailov, Pisma v ZhETF, 30 (1979), 443–448

[4] E. Date, M. Kashiwara, M. Jimbo, T. Miwa, “Transformation groups for soliton equations”, Non-linear Integrable Systems – Classical Theory and Quantum Theory, Proc. of RIMS Symp. (Kyoto, Japan, 13–16 May, 1981), eds. M. Jimbo and T. Miwa, World Scientific, Singapore, 1983, 39–120 | MR

[5] L. A. Dickey, Soliton Equations and Hamiltonian System, World Scientific, Singapore, 1991 | MR | Zbl

[6] N. Ya. Vilenkin, A. U. Klimyk, Representation of Lie Groups and Special Functions. V. 3. Classical and Quantum Groups and Special Functions, Kluwer, Dordrecht, 1992 | MR | Zbl

[7] N. Ya. Vilenkin, A. U. Klimyk, Representation of Lie Groups and Special Functions. Recent Advances, Kluwer, Dordrecht, 1995 | MR

[8] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon, Oxford, 1995 | MR | Zbl

[9] S. C. Milne, “Summation theorems for basic hypergeometric series of Schur function argument”, Progress in Approximation Theory, eds. A. A. Gonchar and E. B. Saff, Springer, New York, 1992, 51–77 | DOI | MR | Zbl

[10] K. I. Gross, D. S. Richards, Trans. Am. Math. Soc., 301 (1987), 781–811 | DOI | MR | Zbl

[11] K. Ueno, K. Takasaki, Adv. Stud. Pure Math., 4, 1984, 1–95 | MR | Zbl

[12] P. Vinternits, A. Yu. Orlov, TMF, 113 (1997), 231–260 | DOI | MR

[13] L. A. Dickey, Mod. Phys. Lett. A, 8 (1993), 1357–1377 | DOI | MR | Zbl

[14] M. Adler, T. Shiota, P. van Moerbeke, Phys. Lett. A, 194 (1994), 33–34 | DOI | MR

[15] A. Yu. Orlov, “Vertex operators, $\partial$ bar problem, symmetries, Hamiltonian and Lagrangian formalism of $(2+1)$ dimensional integrable systems”, Proc. 3rd Kiev Intl. Workshop, V. 1 (1987), eds. V. G. Bar'yakhtar et al., World Scientific, Singapore, 1988, 116–134 | MR | Zbl

[16] K. Takasaki, Commun. Math. Phys., 181 (1996), 131–156 ; T. Nakatsu, K. Takasaki, S. Tsujimaru, Nucl. Phys. B, 443 (1995), 155–197 | DOI | MR | Zbl | DOI | MR | Zbl

[17] G. E. Andrews, E. Onofri, “Lattice gauge theory, orthogonal polynomials, and $q$-hypergeometric functions”, Special Functions{:} Group Theoretical Aspects and Applications, eds. R. A. Askey, T. H. Koornwinder, W. Schempp, Reidel, Dordrecht, 1984, 163–188 | DOI | MR | Zbl

[18] A. Odzijevicz, Commun. Math. Phys., 192 (1998), 183–215 | DOI | MR

[19] A. Yu. Morozov, L. Vinet, Phys. Lett. A, 8 (1993), 2891–2902 | DOI | MR | Zbl

[20] I. Loutsenko, V. Spiridonov, Nucl. Phys. B, 538 (1999), 731–758 | DOI | MR | Zbl

[21] A. Zabrodin, Commuting difference operators with elliptic coefficients from Baxter's vacuum vectors, ; I. Frenkel, V. Turaev, “Elliptic solutions of the Yang–Baxter equations and modular hypergeometric functions”, The Arnold–Gelfand Mathematical Seminars, eds. V. I. Arnold, I. M. Gelfand, V. S. Retakh, M. Smirnov, Birkhauser, Boston, 1997, 171–204 ; A. Zhedanov, V. Spiridonov, Commun. Math. Phys., 210 (2000), 49–83 E-print QA/9912218 | MR | DOI | MR | Zbl | DOI | MR | Zbl

[22] T. Takebe, K. Takasaki, Rev. Math. Phys., 7 (1995), 743–808 ; E-print hep-th/94050096 | DOI | MR | Zbl

[23] M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, Phys. Rev. Lett., 84 (2000), 5106–5109 | DOI

[24] G. Segal, G. Wilson, Publ. Math. IHES, 61 (1985), 5–65 | DOI | Zbl

[25] A. Yu. Orlov, “Volterra operator algebra for zero-curvature representation: Universality of KP”, Nonlinear and Turbulent Processes in Physics (Potsdam–Kiev, 1991), Series in Nonlinear Phys., 126, eds. A. Fokas, D. Kaup, V. E. Zakharov, Springer, Berlin, 1993, 126–131 | MR