Page 286 - Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 271
2
= −2 b b q e p e q ψ p ψ q A ,
pq
p
p,q
S =2R. (6.7.40)
Referring to (6.7.33), (6.7.28), and (6.7.35),
2

T = b p b q b r e p e q e r ∂ θ r
p,q,r ∂e p ∂e q

=6 b p b q e p e q ψ p ψ q b r e r A A qr
pr
p,q r

1
=6 (b p + b q )A pq
2 − ψ p ψ q
b p b q e p e q ψ p ψ q
p,q
2
=6 b b q e p e q ψ p ψ q A pq − 6
p b p e p θ p b q e q θ q
p,q p q
2
= −(3R +6v ).
x
Hence,
2
v xxx = −v t + R +2R − (3R +6v )
x
2
= −(v t +6v ),
x
which completes the verification of the first form of solution of the KdV
equation by means of partial derivatives with respect to the exponential
functions.
6.7.4 The Wronskian Solution
Theorem 6.14. The determinant A in Theorem 6.7.1 can be expressed
in the form
A = k n (e 1 e 2 ··· e n ) 1/2 W,

where k n is independent of x and t, and W is the Wronskian defined as
follows:
j−1
W = D (φ i ) , (6.7.41)

x
n
where
1/2 −1/2
φ i = λ i e + µ i e , (6.7.42)
i i
3
e i = exp(−b i x + b t + ε i ), (6.7.43)
i
1
n
λ i = (b p + b i ),
2
p=1
n

µ i = (b p − b i ). (6.7.44)
p=1
p =i