Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 269

∂

∂em

(A

ij

)=−A

im

A

mj

. (6.7.26)

Let

ψp=

∑

s

A

sp

. (6.7.27)

Then, (6.7.26) can be written

∑

r

brerA

ir

A

rj

=

1

2

(bi+bj)A

ij

−ψiψj. (6.7.28)

From (6.7.27) and (6.7.26),

∂ψp

∂eq

=−A

pq

∑

s

A

sq

=−ψqA

pq

. (6.7.29)

Let

θp=ψ

2

p. (6.7.30)

Then,

∂θp

∂eq

=− 2 ψpψqA

pq

(6.7.31)

=

∂θq

∂ep

, (6.7.32)

∂

2

θr

∂ep∂eq

=− 2

∂

∂ep

(ψqψrA

qr

)

=2(ψpψqA

pr

A

qr

+ψqψrA

qp

A

rp

+ψrψpA

rq

A

pq

),

which is invariant under a permutation ofp,q, andr. Hence, ifGpqris any

function with the same property,

∑

p,q,r

Gpqr

∂

2

θr

∂ep∂eq

=6

∑

p,q,r

GpqrψpψqA

pr

A

qr

. (6.7.33)

The above relations facilitate the evaluation of the derivatives ofvwhich,

from (6.7.7) and (6.7.27) can be written

v=

∑

m

(ψm−bm).

Referring to (6.7.29),

∂v

∂er

=−ψr

∑

m

A

mr

=−ψ

2

r

=−θr. (6.7.34)