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