1. INTRODUCTION
5
was mentioned by Henon in a letter to Flaschka as early as 1973.) Incidentally,
(1.28)-(1.32) illustrate the factorization of the Lax difference expression L alluded
to earlier. The implication (1.34) was first systematically studied by Adler [1] using
factorization techniques. Transformations between the KM and TL systems were
studied earlier by Wadati [87] (see also [84]). In a recent paper [39] the converse of
(1.34) was established. More precisely, assuming the existence of a solution (ai, &i)
of the TLg equations (1.15), that is,
(1.35) TL
p
(ai,6i)=0 ,
a solution p of the KM^ equations (1.16) and another solution (c^fo) of the TL^
equations (1.15) are constructed,
(1.36) KM5(/)) = 0, TLg{a2, fe) = 0
related to each other by the Miura-type transformation (1.31), (1.32) (we refer to
Chapter 7 for a detailed discussion of these facts). Equations (1.34) and especially
(1.35), (1.36) yield the possibility of transferring classes of solutions (such as finite-
gap solutions) from the Toda hierarchy to the KM hierarchy and vice versa.
Having illustrated items (i)-(iii) to some extent, we finally turn to a description
of the content of each chapter. In Chapter 2 we develop an alternative recursive ap-
proach to the Toda hierarchy modeled after Al'ber [6]. In particular, we recursively
compute the difference expressions P2g+2 in (1.13). We chose to develop this ap-
proach in detail since it most naturally leads to the fundamental Burchnall-Chaundy
polynomials and hence to the underlying hyperelliptic curves in connection with the
stationary Toda hierarchy. In addition it provides direct insight into the spectral
properties of the underlying Lax operator as detailed later in Chapter 4.
Chapter 3 is devoted to the algebro-geometric approach to integrate nonlinear
evolution equations and in particular to the Baker-Akhiezer (BA) function, the
fundamental object of this approach. Historically, these techniques go back to
the work of Baker [8], Burchnall and Chaundy [16], [17], and Akhiezer [5]. The
modern approach was initiated by Its and Matveev [46] in connection with the KdV
equation and further developed into a powerful machinery by Krichever (see, e.g.,
the review [58]) and others. We refer, in particular, to the extensive treatments
in [10], [25], [26], [27], [62], [72], and [76]. In the special context of the Toda
equations we refer to [2], [19], [26], [27], [56], [58], [62], [66], and [70]. Our
own presentation starts with commuting difference expressions and their associated
hyperelliptic curves and then develops the stationary algebro-geometric approach
from first principles. In particular, we chose to follow Jacobi's classic representation
of positive divisors of degree g of the hyperelliptic curve (1.21) [47] which was
first applied to the KdV case by Mumford [73], Sect. Ill a).l with subsequent
extensions due to McKean [63]. The reader will find a meticulous account which
provides more details on the BA-function than usually found in the literature (see,
e.g., Theorem 3.5).
Spectral theoretic properties and Green's functions of self-adjoint £2(Z) reali-
zations H of L are the main topic of Chapter 4. Assuming that L is defined in
terms of stationary solutions (a, b) of the TL5 equations, we determine the spec-
trum of the Jacobi operator H in Theorem 4.2 and provide a link between the 2 x 2
spectral matrix of H and our recursive approach to the stationary Toda hierar-
chy (cf. (4.32)-(4.36)). The latter result appears to be new and underscores the
fundamental importance of the recursion formalism chosen in Chapter 2.
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