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 Pn+1,n+2

and apply (6.9.7). To prove (6.9.18), perform the same row operations on

P

∗

n+1,n+2

, apply the third equation in (6.9.8), and transpose the result.

To prove (6.9.19), note that

Q+Qn+2,n+2=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 1 1

c 2 1

··· ···

[bij]n ··· ···

··· ···

cn 1

−c 1 −c 2 ··· −cn 00

− 1 − 1 ··· − 101

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+2

. (6.9.25)

The row operations

R

′

i

=Ri−Rn+2, 1 ≤i≤n,

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 thatQ−Qn+2,n+2can 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

′

i=Ri+Rn+2,^1 ≤i≤n,

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) byF. Then, it is required to prove that

F= 0. Applying (6.9.3), (6.9.5), (6.9.11), and (6.9.17),

Ax=

∑

r

∂A

∂θr

∂θr

∂x

=ω

∑

r

crArr (6.9.26)

=ω

∑

r

∑

s

crArs

=ωPn+1,n+2

=ωQn+1,n+2. (6.9.27)

Taking the complex conjugate of (6.9.27) and referring to (6.9.18),

A

∗

x

=−ωP

∗

n+1,n+2

=(−1)

n

ωQn+2,n+1.