Determinants and Their Applications in Mathematical Physics

5.7 The One-Variable Hirota Operator 225

=D

r

{F(x)+(−1)

r

G(x)},D=

d

dx

. (5.7.18)

Hence,

ψ 2 r=D

2 r

log(fg)

=D

2 r

(φ)

=u 2 r.

Similarly,

ψ 2 r+1=u 2 r+1.

Hence,ψr=urfor all values ofr.

In (5.7.17), put

Hi=H

i

(e

F

,e

G

),

so that

H 0 =e

F+G

and put

aij=

(

i− 1

j

)

ψi−j,j<i,

aii=− 1.

Then,

ai 0 =ψi=ui

and (5.7.17) becomes

i

∑

j=0

aijHj=0,i≥ 1 ,

which can be expressed in the form

i

∑

j=1

aijHj=−ai 0 H 0

=−e

F+G

ui,i≥ 1. (5.7.19)

This triangular system of equations in theHj is similar in form to the

triangular system in Section 2.3.5 on Cramer’s formula. The solution of

that system is given in terms of a Hessenbergian. Hence, the solution of

(5.7.19) is also expressible in terms of a Hessenbergian,

Hj=e

F+G

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

u 1 − 1

u 2 u 1 − 1

u 3 2 u 2 u 1 − 1

u 4 3 u 3 3 u 2 u 1 − 1

.......................

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,