In the first part of this paper we consider the transformation of the cubic identities for general Korteweg–de Vries (KdV) tau functions from [Mishev, J. Math. Phys. 40, 2419–2428 (1999)] to the specific identities for trigonometric KdV tau functions. Afterwards, we consider the Fay identity as a functional equation and provide a wide set of solutions of this equation. The main result of this paper is Theorem 3.4, where we generalize the identities from Mishev. An open problem is the transformation of the cubic identities from Mishev to the specific identities for elliptic KdV tau functions.
REFERENCES
1.
Adler
, M.
and van Moerbeke
, P.
, “A matrix integral solution to two-dimensional -gravity
,” Commun. Math. Phys.
147
, 25
–56
(1992
).2.
Fay
, J.
, Theta Functions on Riemann Surfaces
, Lecture Notes in Mathematics
Vol. 352
(Springer
, Berlin, 1973
).3.
Mishev
, Y. P.
, “On the Fay identity for KdV tau functions and the identity for the Wronskian of squared solutions of Sturm-Liouville equation
,” J. Math. Phys.
40
, 2419
–2428
(1999
); QA/9804077.4.
Mumford
, D.
, Tata Lectures on Theta
I, II, Progress in Mathematics Vols. 28, 43 (Birkhäuser
, Boston, Inc., 1983–1984
).5.
Ohta
, Y.
, Satsuma
, J.
, Takahashi
, D.
, and Tokihiro
, T.
, “An elementary introduction to Sato theory
,” Prog. Theor. Phys. Suppl.
94
, 210
–241
(1988
).6.
Shiota
, T.
, “Characterization of Jacobian varieties in terms of soliton equations
,” Invent. Math.
83
, 333
–382
(1986
).7.
Shiota
, T.
(private communication).8.
Takasaki
, K.
and Takebe
, T.
, “Integrable hierarchies and dispersionless limit
,” Rev. Math. Phys.
7
, 743
–808
(1995
).© 2006 American Institute of Physics.
2006
American Institute of Physics
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