8 2. TH E TODA HIERARCHY
In connection with (2.4) we define the diagonal and upper and lower triangular
parts of R as follows
D ro ( \\ u( \ JR(rn,m), m = n
R0 = {ito(m,n)}m5nGZ, Ro{m,n) =
(p(ra,ra) ,
i,n)
=
i , n )
=
;
\o,
(2.6) = {P±(m,n)}
m
,
n e Z
, ,R±(m,n) = ^ v ; ' ; /
10, otherwise
Clearly,
(2.7) R = R+ + R0 + R-.
Given these notations one can now introduce the Toda hierarchy. Let
a(t) = {a(n,t)}nez G *°°(Z), b(t) = {b(n,t)}neZ G £°°(Z), t G R,
^ ' ' 0 ^ a(n,.), 6(n,.) G C^M), n G Z
and introduce the difference expressions (L(£),P2p+2(£)) (the Lax pair) in £°°(Z)
(2.9) L(*) = a(t)S+ + a~(t)5" - 6(*), t G R,
P2p+2(t) = -L(ty^ + £ , ( * ) + 2a(t)/
J
(05 + ]L(t)^^' + fg+1(t),
g G N0, t G R,
where {/j(rM)}ojs+i and {ft(ft,£)}ojs satisfy the recursion relations
/o = 1, 9o = - c i ,
(2.11) 2/
i +
i + ft + ?" + 2bfj = 0, 0 j ft
Pi+i - ^"+i +
2
l
a 2
/ / "
(tt~)2/~]
+ % ; " 9j] = 0, 0 j 9 ~ 1
Note that a enters in fj and ft only quadratically. Then the Lax equation
(2.12) jtL{t) - [P2g+2{t), L(t)} = 0, t G R
(here [., .] denotes the commutator) is equivalent to
TLfl(a(t),6(t))i = d(*) + a(t)[g+(t) + gg(t) + f++1(t) + fg+1(t)
+ 2b+(t)f+(t)) = 0,
(2.13)
TL9(o(t),6(t))2 = b(t) + 2{b(t)(gg(t) + fg+1(t)) + a(t)2f+(t)
9
a-(t)2f-(t) + b(t)2fg(t)]=0, teR.
Varying g No yields the Toda hierarchy
(2.14) TL3(a, b) = (TLff (a, b)x, TL9(a,
b)2)T
= 0, g N0.
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