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 byH

n

, is defined as

follows: Iff=f(x) andg=g(x), then

H

n

(f, g)=

[(

∂

∂x

−

∂

∂x

′

)n

f(x)g(x

′

)

]

x′=x

=

∑n

r=0

(−1)

r

(

n

r

)

D

n−r

(f)D

r

(g),D=

d

dx

. (5.7.1)

The factor (−1)
r
distinguishes this sum from the Leibnitz formula for

D
n
(fg). The notationHx,Hxx, etc., is convenient in some applications.

Examples.

Hx(f, g)=H

1

(f, g)=fxg−fgx

=−Hx(g, f),

Hxx(f, g)=H

2

(f, g)=fxxg− 2 fxgx+fgxx

=Hxx(f, g).

Lemma.

e

zH

(f, g)=f(x+z)g(x−z).

Proof. Using the notationr=i(→j) defined in Appendix A.1,

e

zH

(f, g)=

∞

∑

n=0

z

n

n!

H

n

(f, g)

=

∞

∑

n=0

z

n

n!

n(→∞)

∑

r=0

(−1)

r

(

n

r

)

D

n−r

(f)D

r

(g)

=

∞

∑

r=0

(−1)

r

D

r

(g)

r!

∞

∑

n=0(→r)

z

n

D

n−r

(f)

(n−r)!

(puts=n−r)

=

∞

∑

r=0

(−1)

r

z

r

D

r

(g)

r!

∞

∑

s=0

z

s

D

s

(f)

s!

.

These sums are Taylor expansions ofg(x−z) andf(x+z), respectively,

which proves the lemma.