Page 289 - Determinants and Their Applications in Mathematical Physics

274 6. Applications of Determinants in Mathematical Physics

into the KdV equation yields
F

, (6.7.56)
w 2
u t +6uu x + u xxx =2D x
where
F = ww xt − w x w t +3w 2 − 4w x w xxx + ww xxxx .
xx
Hence, the KdV equation will be satisfied if
F =0. (6.7.57)

Theorem 6.15. The KdV equation in the form (6.7.56) and (6.7.57) is
satisfied by the Wronskian w defined as follows:
j−1
w = D (ψ i ) ,

x
n
where
1 2
ψ i = exp b z φ i ,
4 i
1/2 −1/2
φ i = p i e + q i e ,
i i
3
e i = exp(−b i x + b t).
i
z is independent of x and t but is otherwise arbitrary. b i , p i , and q i are
constants.
When z =0, p i = λ i , and q i = µ i , then ψ i = φ i and w = W so that this
theorem differs little from Theorem 6.14 but the proof of Theorem 6.15
which follows is direct and independent of the proofs of Theorems 6.13 and
6.14. It uses the column vector notation and applies the Jacobi identity.
Proof. Since
3
(D t +4D )φ i =0,
x
it follows that
3
(D t +4D )ψ i =0. (6.7.58)
x
Also
2
(D z − D )ψ i =0. (6.7.59)
x
1 2
Since each row of w contains the factor exp b z ,
4 i
w = e Bz W,
where
j−1
W = D (φ i )

x
n
and is independent of z and
2
1
B = b .
4 i
i