Page 240 - Determinants and Their Applications in Mathematical Physics

5.7 The One-Variable Hirota Operator 225
d
= D {F(x)+(−1) G(x)}, D = . (5.7.18)
r
r
dx
Hence,
ψ 2r = D 2r log(fg)
= D (φ)
2r
= u 2r .
Similarly,
ψ 2r+1 = u 2r+1 .
Hence, ψ r = u r for all values of r.
In (5.7.17), put
H i = H (e ,e ),
G
F
i
so that
H 0 = e F +G
and put

i − 1
a ij = ψ i−j , j < i,
j
a ii = −1.
Then,

and (5.7.17) becomes a i0 = ψ i = u i

i

a ij H j =0, i ≥ 1,
j=0
which can be expressed in the form

i

a ij H j = −a i0 H 0
j=1
= −e F +G u i , i ≥ 1. (5.7.19)
This triangular system of equations in the H j 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,

u 1 −1

u 2 u 1 −1
H j = e F +G u 3 2u 2 u 1 −1 ,

u 4 3u 3 3u 2 u 1 −1
.......................

n