Quasi-periodic solutions of the orthogonal KP equation. Translation groups for soliton equations. V
type: Item
instance of: scholarly article
title: Quasi-periodic solutions of the orthogonal KP equation. Translation groups for soliton equations. V
zbMATH Open document ID: 0571.35101
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1982-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14L10
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3910947
zbMATH Keywords: quasi-periodic solutions
zbMATH Keywords: BKP hierarchy
zbMATH Keywords: quasi-periodic \(\tau \) - functions
zbMATH Keywords: theta functions
zbMATH Keywords: Prym varieties of algebraic curves
zbMATH Keywords: wave functions
zbMATH Keywords: soliton solutions
zbMATH Keywords: Abelian integrals
zbMATH Keywords: pole divisor
zbMATH Keywords: Riemann's theta function
author: E. Date
author: M. Jimbo
author: Masaki Kashiwara
author: T. Miwa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195183298
OpenAlex ID: W1552007708
cites work: Transformation groups of soliton equations. IV: A new hierarchy of soliton equations of KP-type
cites work: Theta functions on Riemann surfaces
DOI: 10.2977/PRIMS/1195183298
description: scientific article; zbMATH DE number 3910947