1. INTRODUCTION
3
Since the literature on the Toda lattice (even if one considers only the infinite
lattice Z, our main interest) is extensive, we will only refer to a few standard
monographs such as [29], [76], [77], [84]. In the case of the Kac-van Moerbeke
system we refer to [15], [39], [49], [50], [53], [60], [61], [62], [71], [87] and the
references therein.
While (1.5) and (1.6) describe the original Toda and Kac-van Moerbeke lattices,
one can develop a systematic generalization to Lax pairs of the type (L, -P2P+2)
and (M, (52^+2), where P22+2 (Q2P+2) are (matrix-valued) difference expressions of
order 2g + 2 with certain polynomial coefficients in a, b (p). The associated Lax
equations
(1.13) L-[P2g+2,L}=0
and
(1.14) M-[Q2g+2,M} = 0
(cf. Chapter 2) are then equivalent to the TL5 and KMP equations denoted by
(1.15) TL^(a,6) = 0
and
(1.16) KM3(p) = 0.
Varying g G NQ then yields the corresponding hierarchies of nonlinear evolution
equations for (a, b) and p.
The special case of stationary TL5 and KM5 equations, characterized by L = 0,
M 0 in (1.13), (1.14), or equivalently, by commuting difference expressions of the
type
(1.17) [P2g+uL]=0,
(1.18) [Q2g+2,M]=0,
then yields a polynomial relationship between L and P23+2, respectively M and
325+2- In fact, (1.17) and (1.18) imply the following analogs of the Burchnall-
Chaundy polynomials familiar from the theory of commuting ordinary differential
expressions [16], [17]
(1.19)
ra=0
2?+l
(1.20) Qi
p + 2
= f l
(M2
- c
m
), {em}om25+i C C.
m=0
In particular, (1.19) and (1.20) yield the following hyperelliptic curves
2H-1
(1.21) y2= ! ! ( * - £ ) ,
m=0
2
5
+l
(1.22)
y2=
lJ(w
2
-e
m
),
m=0
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