Quasi-periodic solutions of the orthogonal KP equation. Translation groups for soliton equations. V
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.
scientific article; zbMATH DE number 3910947