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± = {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.