4 1. INTRODUCTION
the fundamental ingredients of the algebro-geometric approach for the TL and KM
hierarchies. For spectral theoretic reasons (see, e.g., Theorems 4.2 and 8.2) algebro-
geometric solutions (a, b) and p of (1.15) and (1.16) are called #-gap solutions
following the conventional terminology.
If 6 denotes the Riemann theta function associated with the curve (1.21) (and
a fixed homology basis), the ultimate goal of the algebro-geometric approach is
then a ^-function representation of the solutions (a, b) and p of the TL
r
and KMr
equations,
(1.23) TLr(a, b) = 0, KMr(p) = 0 , r e N0
with #-gap initial conditions,
(1.24) (a,6) =
(a(°\&0)),
p =
p(0)
at t = *0,
where (a^°\b^) and p^ are stationary solutions of the TL5 and KM5 equations,
that is,
TLfl(a°\6°)=0, a°=6°=0,
(1.25) KMfl(p(°) = 0, /
( 0 )
=0
for some fixed j 6 N o .
Next, we illustrate the close connection between the Toda hierarchy and its
modified counterpart, the KM hierarchy. Introducing the difference expressions
(1.26) A = PoS+ + pe, A* = p-S~ + pe
in ^°°(Z) one infers that
(1.27) M =
0 A*
A 0
(1-28)
M 2 =
( V AA*)
=Ll0£2
'
with
(1.29) Ll=A*A, L2 = AA\
(1.30) Lk = akS+ + alS~ - bk, k = 1,2,
(1-31) ai = pep0, h = -pi -
(p~)2,
(1-32) a2=ptPo, b2 =
-p2e-p20
and
(1.33) Q
2 9 + 2
=
(Pl'209+2
p2 2
°
9 +
J = Plt2g+2 0 P2,2S+2
Here Pk,2g+2 is constructed as in (1.10) and (1.13) with (a, 6) replaced by (afc,6fc),
fc = 1,2, respectively. Relations (1.28) and (1.33) then can be exploited to prove
the implication
(1.34) KM5(/) = 0 = TL9{ak,bk) - 0, k = 1,2,
that is, a solution p of the KM9 equations (1.16) yields two solutions (afc,6fc),
A: = 1,2 of the TL^ equations (1.15) related to one another by (1.31), (1.32), the
discrete analog of Miura's transformation [67], familiar from the (m)KdV hierarchy.
(According to a footnote in [71], the connection between the KM and TL lattices
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