Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 275

Hence,wwzz−w
2
z

=0,

F=wwxt−wxwt+3w

2

xx

− 4 wxwxxx+wwxxxx+3(wwzz−w

2

z

)

=w

[

(wt+4wxxx)x−3(wxxxx−wzz)

]

−wx(wt+4wxxx)+3(w

2

xx

−w

2

z

). (6.7.60)

The evaluation of the derivatives of a Wronskian is facilitated by expressing

it in column vector notation.

Let

W=

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 Cn− 2 Cn− 1

∣

∣

n

, (6.7.61)

where

Cj=

[

D

j

x(ψ^1 )D

j

x(ψ^2 )···D

j

x(ψn)

]T

.

The significance of the row of dots above the (n−3) columnsC 0 toCn− 4

will emerge shortly. It follows from (6.7.58) and (6.7.59) that

Dx(Cj)=Cj+1,

Dz(Cj)=D

2

x

(Cj)=Cj+2,

Dt(Cj)=− 4 D

3

x

(Cj)=− 4 Cj+3. (6.7.62)

Hence, differentiating (6.7.61) and discarding determinants with two

identical columns,

wx=

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 Cn− 2 Cn

∣

∣

n

,

wxx=

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 Cn− 1 Cn

∣

∣

n

+

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 Cn− 2 Cn+1

∣

∣

n

,

wz=

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 CnCn− 1

∣

∣

n

+

∣

∣

..............................

C 0 C 1 ···Cn− 4 Cn− 3 Cn− 2 Cn+1

∣

∣

n

,

etc. The significance of the row of dots above columnsC 0 toCn− 4 is

beginning to emerge. These columns are common to all the determinants

which arise in all the derivatives which appear in the second line of (6.7.60).

They can therefore be omitted without causing confusion.

Let

Vpqr=

∣

∣

..............................

C 0 C 1 ···Cn− 4 CpCqCr

∣

∣

n

. (6.7.63)

Then,Vpqr=0ifp,q, andrare not distinct andVqpr=−Vpqr, etc. In this

notation,

w=Vn− 3 ,n− 2 ,n− 1 ,

wx=Vn− 3 ,n− 2 ,n,

wxx=Vn− 3 ,n− 1 ,n+Vn− 3 ,n− 2 ,n+1,

wxxx=Vn− 2 ,n− 1 ,n+2Vn− 3 ,n− 1 ,n+1+Vn− 3 ,n− 2 ,n+2,