Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 273

Hence, from (6.7.48),

K

(n)

ij

Un

=

μj

λi

[

H

(n)

ij

Xn

]

yi=−xi=bi

=

μj

λi

∏n

p=1

(bp+bj)

(bi+bj)

∏n

p=1

p=i

(bp−bj)

=

2 λjμj

(bi+bj)λiμi

. (6.7.50)

Hence,

|Eij|n=

∣

∣

∣

∣

δijei+

2 λjμj

(bi+bj)λiμi

∣

∣

∣

∣

n

. (6.7.51)

Multiply rowiof this determinant byλiμi,1≤i≤n, and divide column

jbyλjμj,1≤j≤n. These operations do not affect the diagonal elements

or the value of the determinant but now

|Eij|n=

∣

∣

∣

∣

δijei+

2

bi+bj

∣

∣

∣

∣

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 ···en)

1 / 2

W=UnA, (6.7.53)

which completes the proof of the theorem sinceUnis independent ofxand

t.

It follows that

logA= constant +

1

2

∑

i

(−bix+b

3

it) + logW. (6.7.54)

Hence,

u=2D

2

x

(logA)=2D

2

x

(logW) (6.7.55)

so that solutions containingAandW have equally valid claims to be

determinantal solutions of the KdV equation.

6.7.5 Direct Verification of the Wronskian Solution

The substitution

u=2D

2

x

(logw)