Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 271

=− 2

∑

p,q

b

2

pbqepeqψpψqA

pq

,

S=2R. (6.7.40)

Referring to (6.7.33), (6.7.28), and (6.7.35),

T=

∑

p,q,r

bpbqbrepeqer

∂

2

θr

∂ep∂eq

=6

∑

p,q

bpbqepeqψpψq

∑

r

brerA

pr

A

qr

=6

∑

p,q

bpbqepeqψpψq

[

1

2

(bp+bq)A

pq

−ψpψq

]

=6

∑

p,q

b

2

p

bqepeqψpψqA

pq

− 6

∑

p

bpepθp

∑

q

bqeqθq

=−(3R+6v

2

x).

Hence,

vxxx=−vt+R+2R−(3R+6v

2

x

)

=−(vt+6v

2

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 determinantAin Theorem 6.7.1 can be expressed

in the form

A=kn(e 1 e 2 ···en)

1 / 2

W,

whereknis independent ofxandt, andW is the Wronskian defined as

follows:

W=

∣

∣Dj−^1

x

(φi)

∣

∣

n

, (6.7.41)

where

φi=λie

1 / 2

i +μie

− 1 / 2

i , (6.7.42)

ei= exp(−bix+b

3

i

t+εi), (6.7.43)

λi=

1

2

n

∏

p=1

(bp+bi),

μi=

n

∏

p=1

p=i

(bp−bi). (6.7.44)