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Quasi-periodic solutions of the orthogonal KP equation. Translation groups for soliton equations. V. (English) Zbl 0571.35101

In this note the authors study quasi-periodic solutions of the BKP hierarchy introduced in part II of this article [Proc. Japan Acad., Ser. A 57, 387-392 (1981; Zbl 0538.35066)]. Their main result is the theorem which states that quasi-periodic \(\tau\)-functions for the BKP hierarchy are the theta functions on the Prym varieties of algebraic curves admitting involutions with two fixed points. Section 1 is devoted to the study of wave functions associated with soliton solutions. In Section 2, quasi-periodic wave functions for the BKP hierarchy were constructed by using the theory of Abelian integrals. An explicit formula was given in terms of the theta functions on the Prym varieties.
A derivation is given through the examination of the geometrical properties of the wave functions associated with soliton solutions. The authors found that the pole divisor of the wave function belongs to a translation of the Prym variety which is tangent to the theta divisor in the Jacobian variety. On the other hand, the quasi-periodic BKP \(\tau\)- function must be the square root of Riemann’s theta function. If the relevant curve has an involution with two fixed points, then Riemann’s theta function is known to reduce to the square of the theta function on the associated Prym variety when its arguments are restricted to a translation of the Prym part. This fact agrees with the observation in this paper for the soliton case.
Reviewer: L.-Y.Shih

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
14H05 Algebraic functions and function fields in algebraic geometry
14L10 Group varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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[1] Date, E., Kashiwara, M. and Miwa, T., Proc. Japan Acad. 57 A (1981) 387.
[2] Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., A new hierarchy of soliton equa- tions of KP-type, Physica 4D (1982) 343. · Zbl 0571.35100
[3] 185.
[4] Fay, J. D., Theta Functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer, 1973. · Zbl 0281.30013 · doi:10.1007/BFb0060090
[5] Van Moerbecke, P. and Mumford, D., Acta Math. 143 (1979), 93.
[6] Adler, M. and van Moerbecke, P., Advances in Math. 38 (1980), 267, 318.
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