Page 238 - Determinants and Their Applications in Mathematical Physics

5.7 The One-Variable Hirota Operator 223

Proof. First proof (Caudrey). The Hessenbergian satisfies the recurrence
relation (Section 4.6)

n
n
E n+1 = u r+1 E n−r . (5.7.7)
r
r=0
Let
H (f, g)
n
F n = , f = f(x),g = g(x),F 0 =1. (5.7.8)
fg
The theorem will be proved by showing that F n satisfies the same
recurrence relation as E n and has the same initial values.
Let
zH
 (f,g)
e

 fg

n n
 ∞ !
 H (f,g)
z
K = n! fg (5.7.9)
 n=0
n
 ∞ !

z F n
 .

n!
n=0
Then,
∞
∂K z n−1
= F n (5.7.10)
∂z (n − 1)!
n=1
∞
n
z F n+1
= . (5.7.11)
n!
n=0
From the lemma and (5.7.6),
f(x + z)g(x − z) 1
K = = exp 2 {φ(x + z)+ φ(x − z)
f(x)g(x)

+ψ(x + z) − ψ(x − z) − 2φ(x)} . (5.7.12)
Differentiate with respect to z, refer to (5.7.11), note that
D z (φ(x − z)) = −D x (φ(x − z))
etc., and apply the Taylor relations (5.7.2) from the previous section. The
result is
∞
z F n+1 1
n
= D {φ(x + z) − φ(x − z)+ ψ(x + z)+ ψ(x − z)} K
n! 2
n=0

∞ 2n+1 2n+2 ∞ 2n+1
2n
z D (φ) z D (ψ)
= + K
(2n + 1)! (2n)!
n=0 n=0

∞ 2n+1 ∞
2n
z u 2n+2 z u 2n+1
= + K
(2n + 1)! (2n)!
n=0 n=0