Page 288 - Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 273

Hence, from (6.7.48),
(n)
(n)
K H
=
ij µ j ij
X n
U n
λ i
y i =−x i =b i
(b p + b j )
n
p=1
= µ j
(b i + b j ) n (b p − b j )
λ i
p=1
p =i
= (b i + b j )λ i µ i . (6.7.50)
2λ j µ j
Hence,

|E ij | n = δ ij e i + 2λ j µ j . (6.7.51)

(b i + b j )λ i µ i n

Multiply row i of this determinant by λ i µ i ,1 ≤ i ≤ n, and divide column
j by λ j µ j ,1 ≤ j ≤ n. These operations do not affect the diagonal elements
or the value of the determinant but now

2

|E ij | n = δ ij e i +

b i + b j n

= A. (6.7.52)
It follows from (6.7.46) and (6.7.49) that
2 n(n−1)/2 (e 1 e 2 ··· e n ) 1/2 W = U n A, (6.7.53)
which completes the proof of the theorem since U n is independent of x and
t.
It follows that
1 3
log A = constant + (−b i x + b t) + log W. (6.7.54)
2 i
i
Hence,

2
2
u =2D (log A)=2D (log W) (6.7.55)
x x
so that solutions containing A and W have equally valid claims to be
determinantal solutions of the KdV equation.

6.7.5 Direct Verification of the Wronskian Solution
The substitution
2
u =2D (log w)
x