5.7 The One-Variable Hirota Operator 223
Proof. First proof(Caudrey). The Hessenbergian satisfies the recurrence
relation (Section 4.6)
En+1=
n
∑
r=0
(
n
r
)
ur+1En−r. (5.7.7)
Let
Fn=
H
n
(f, g)
fg
,f=f(x),g=g(x),F 0 =1. (5.7.8)
The theorem will be proved by showing that Fn satisfies the same
recurrence relation asEnand has the same initial values.
Let
K=
e
zH
(f,g)
fg
∑∞
n=0
z
n
n!
H
n
(f,g)
fg
∑∞
n=0
z
n
Fn
n!
.
(5.7.9)
Then,
∂K
∂z
=
∞
∑
n=1
z
n− 1
Fn
(n−1)!
(5.7.10)
=
∞
∑
n=0
z
n
Fn+1
n!
. (5.7.11)
From the lemma and (5.7.6),
K=
f(x+z)g(x−z)
f(x)g(x)
= exp
[
1
2
{φ(x+z)+φ(x−z)
+ψ(x+z)−ψ(x−z)− 2 φ(x)}
]
. (5.7.12)
Differentiate with respect toz, refer to (5.7.11), note that
Dz(φ(x−z)) =−Dx(φ(x−z))
etc., and apply the Taylor relations (5.7.2) from the previous section. The
result is
∞
∑
n=0
z
n
Fn+1
n!
=D
[
1
2
{φ(x+z)−φ(x−z)+ψ(x+z)+ψ(x−z)}
]
K
=
[
∞
∑
n=0
z
2 n+1
D
2 n+2
(φ)
(2n+ 1)!
+
∞
∑
n=0
z
2 n
D
2 n+1
(ψ)
(2n)!
]
K
=
[
∞
∑
n=0
z
2 n+1
u 2 n+2
(2n+ 1)!
+
∞
∑
n=0
z
2 n
u 2 n+1
(2n)!