Page 284 - Determinants and Their Applications in Mathematical Physics
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6.7 The Korteweg–de Vries Equation 269
∂
(A )= −A im A mj . (6.7.26)
ij
Let ∂e m
ψ p = A . (6.7.27)
sp
s
Then, (6.7.26) can be written
1
b r e r A A rj = (b i + b j )A − ψ i ψ j . (6.7.28)
ij
ir
2
r
From (6.7.27) and (6.7.26),
= −A pq A sq
∂ψ p
∂e q
s
= −ψ q A . (6.7.29)
pq
Let
2
θ p = ψ . (6.7.30)
p
Then,
= −2ψ p ψ q A pq (6.7.31)
∂θ p
∂e q
= ∂θ q , (6.7.32)
∂e p
2 ∂
= −2 ∂e p (ψ q ψ r A )
∂ θ r
qr
∂e p ∂e q
=2(ψ p ψ q A A qr + ψ q ψ r A A rp + ψ r ψ p A A ),
pq
rq
pr
qp
which is invariant under a permutation of p, q, and r. Hence, if G pqr is any
function with the same property,
2
=6 G pqr ψ p ψ q A A . (6.7.33)
∂ θ r
pr
qr
G pqr
p,q,r ∂e p ∂e q p,q,r
The above relations facilitate the evaluation of the derivatives of v which,
from (6.7.7) and (6.7.27) can be written
v = (ψ m − b m ).
m
Referring to (6.7.29),
∂v
A mr
= −ψ r
∂e r
m
= −ψ 2
r
= −θ r . (6.7.34)