Page 236 - Determinants and Their Applications in Mathematical Physics

5.7 The One-Variable Hirota Operator 221
5.7 The One-Variable Hirota Operator

5.7.1 Definition and Taylor Relations
Several nonlinear equations of mathematical physics, including the Korteweg–
de Vries, Kadomtsev–Petviashvili, Boussinesq, and Toda equations, can be
expressed neatly in terms of multivariable Hirota operators. The ability of
an equation to be expressible in Hirota form is an important factor in the
investigation of its integrability.
The one-variable Hirota operator, denoted here by H , is defined as
n
follows: If f = f(x) and g = g(x), then

n
∂ ∂

H (f, g)= − f(x)g(x )
n
∂x ∂x
x =x

n
n d
= (−1) r D n−r (f)D (g), D = . (5.7.1)
r
r dx
r=0
The factor (−1) distinguishes this sum from the Leibnitz formula for
r
D (fg). The notation H x , H xx , etc., is convenient in some applications.
n
Examples.
1
H x (f, g)= H (f, g)= f x g − fg x
= −H x (g, f),
2
H xx (f, g)= H (f, g)= f xx g − 2f x g x + fg xx
= H xx (f, g).
Lemma.
e zH (f, g)= f(x + z)g(x − z).
Proof. Using the notation r = i (→ j) defined in Appendix A.1,
∞
z n
e zH (f, g)= H (f, g)
n
n!
n=0
n(→∞)
∞
z n n
= (−1) r D n−r (f)D (g)
r
n! r
n=0 r=0
∞ ∞
r
r
n
(−1) D (g) z D n−r (f)
= (put s = n − r)
r! (n − r)!
r=0 n=0(→r)
∞ ∞
s
s
r
r r
(−1) z D (g) z D (f)
= .
r! s!
r=0 s=0
These sums are Taylor expansions of g(x − z) and f(x + z), respectively,
which proves the lemma.