CHAPTER 1

Introduction

The primary goal of this exposition is to construct all real-valued algebro-

geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke (KM) hier-

archy of nonlinear evolution equations.

While there exists a direct method to construct the finite-gap solutions of the

KM hierarchy, we shall use an alternative route that exploits the close connec-

tion between the Toda and KM hierarchies and characterizes the KM hierarchy

as the modified Toda hierarchy in precisely the same manner that connects the

Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies or

more generally, the Gel'fand-Dickey (GD) hierarchy and its modified counterpart,

the Drinfeld-Sokolov (DS) hierarchy. The deep connection between these hierar-

chies of nonlinear evolution equations and their modified counterparts is based on

Miura-type transformations which in turn rely on factorization techniques of the

associated Lax differential, respectively difference expressions, as will be indicated

below. (Alternatively, one can use the discrete analog of the formal pseudo diffe-

rential calculus in connection with the GD and DS hierarchy as in [60], Ch. IV.)

Accordingly, our approach consists of three main parts:

(i) A thorough treatment of the Toda hierarchy.

(ii) The algebro-geometric approach to completely integrable nonlinear evolution

equations.

(iii) A transfer of classes of solutions of the Toda hierarchy to that of the Kac-van

Moerbeke hierarchy and vice versa.

Our major results then may be summarized as follows:

(a) Construction of an alternative approach to the Toda hierarchy, modeled after

Al'ber [6], Jacobi [47], McKean [63], and Mumford [73], Sect. Ill a).l, particularly

suited to derive its algebro-geometric quasi-periodic finite-gap solutions. Derivation

of an intimate connection of this approach with spectral properties of the corre-

sponding Lax operator.

(/?) A complete presentation of the algebro-geometric approach to the Toda hier-

archy which goes beyond results in the literature and leads, in particular, to an

alternative theta function representation of 6(n,£) (in Flaschka's variables [35],

cf. (6.66)).

(7) A complete derivation of all real-valued algebro-geometric quasi-periodic finite-

gap solutions of the KM hierarchy, our principal new result.

Before we describe the content of each chapter, and hence (a)-(^y) in some

detail, we shall comment on items (i)-(iii) a bit further.

Received by the editor July 2, 1995

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