2 1. INTRODUCTION
The (Abelian) Toda lattice (TL) in its original variables reads
d2
(1.1) ^ Q ( n , t) = exp[Q(n - 1, t) - Q{n, t)] - exp[Q(n, t) - Q(n + 1,*)],
( n , t ) e Z x R
and similarly, the original Kac-van Moerbeke system, also called the Volterra sy-
stem, in physical variables, is of the type
(1.2) —R(n, t) = -{exp[-R(n - 1, t)] - exp[-R(n + 1, £)]}, (n, t j e Z x R ,
at 2
In Flaschka's variables [35] for (1.1) and similarly for (1.2),
a(n,t) = ^ e x p { [ Q ( M ) - Q(n + l,t)]/2},
6(M) = Q(M)/2 ,
6
(
n
)G {+1,-1},
(1.4) p(n,t) = ^ e x p [ -
J
R ( n , t ) / 2 ] , e(n)G {+1,-1}
one can rewrite (1.1) and (1.2) in the form
and
(1.6) KM0(p) =
p-p[(p+)2-(p-)2}=0,
the latter also known as Langmuir lattice. Here "'" denotes d/dt and we employed
the notation f±(n) = f(n ± 1), n G Z and regarded all equations in the multiplica-
tion algebra of sequences. Moreover, introducing the shift operators
(1.7) (S±f)(n) = f(n±l) = f±(n), n Z
in €°°(Z), the systems (1.5) and (1.6) are well-known to be equivalent to the Lax
equations
(1.8) L-[P2,L]=0
and
(1.9) M-[Q2,M}=0.
Here L and P2 are the difference expressions
(1.10) L = aS+ +a~S~ -b, P2 = aS+-a-S~
defined on f°°(Z), and M and Q2 the matrix-valued difference expressions
0 p-S~+pe
\p0S+ +pe 0
( i n ) ; /
n
_
(PePoS+
~ Pep0S 0
^ V °
p+PoS+-pePoS-
defined on £°°(Z) 0
C2,
with pe and p0 the "even" and "odd" parts of p, that is,
(1.12) pe(n, t) = p(2n, t), pQ{n, t) = p(2n + 1, t), n G Z,
assuming a, b, p e £°°(Z).
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