Page 300 - Determinants and Their Applications in Mathematical Physics

6.9 The Benjamin–Ono Equation 285

To prove (6.9.17), perform the row operations (6.9.24) on P n+1,n+2
and apply (6.9.7). To prove (6.9.18), perform the same row operations on
P ∗ , apply the third equation in (6.9.8), and transpose the result.
n+1,n+2
To prove (6.9.19), note that

c 1
1

c 2
1

··· ···

. (6.9.25)
[b ij ] n ··· ···
Q + Q n+2,n+2 =

··· ···

c n 1
0 0

−c 1 − c 2 ··· − c n
−1 − 1 ··· − 1 0 1

n+2
The row operations

R = R i − R n+2 , 1 ≤ i ≤ n,
i
leave a single nonzero element in the last column. The result appears after
applying the second equation in (6.9.7).
To prove (6.9.20), note that Q − Q n+2,n+2 can be expressed as a deter-
minant similar to (6.9.25) but with the element 1 in the bottom right-hand
corner replaced by −1. The row operations

R = R i + R n+2 , 1 ≤ i ≤ n,
i
leave a single nonzero element in the last column. The result appears after
applying the second equation of (6.9.8) and transposing the result.
6.9.3 Proof of the Main Theorem
Denote the left-hand side of (6.9.1) by F. Then, it is required to prove that
F = 0. Applying (6.9.3), (6.9.5), (6.9.11), and (6.9.17),

A x = ∂A ∂θ r
∂θ r ∂x
r

= ω c r A rr (6.9.26)
r

= ω c r A rs
r s
= ωP n+1,n+2
= ωQ n+1,n+2 . (6.9.27)
Taking the complex conjugate of (6.9.27) and referring to (6.9.18),

A = −ωP n+1,n+2
∗
∗
x
=(−1) ωQ n+2,n+1 .
n