Determinants and Their Applications in Mathematical Physics

274 6. Applications of Determinants in Mathematical Physics

into the KdV equation yields

ut+6uux+uxxx=2Dx

(

F

w

2

)

, (6.7.56)

where

F=wwxt−wxwt+3w

2

xx−^4 wxwxxx+wwxxxx.

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 Wronskianwdefined as follows:

w=

∣

∣Dj−^1

x (ψi)

∣

∣

n

,

where

ψi= exp

(

1

4

b

2

iz

)

φi,

φi=pie

1 / 2

i +qie

− 1 / 2

i ,

ei= exp(−bix+b

3

it).

zis independent ofxandtbut is otherwise arbitrary.bi,pi, andqiare

constants.

Whenz=0,pi=λi, andqi=μi, thenψi=φiandw=Wso 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

(Dt+4D

3

x)φi=0,

it follows that

(Dt+4D

3

x)ψi=0. (6.7.58)

Also

(Dz−D

2

x

)ψi=0. (6.7.59)

Since each row ofwcontains the factor exp

(

1

4

b

2

i

z

)

,

w=e

Bz

W,

where

W=

∣

∣Dj−^1

x

(φi)

∣

∣

n

and is independent ofzand

B=

1

4

∑

i

b

2

i

.