Determinants and Their Applications in Mathematical Physics

280 6. Applications of Determinants in Mathematical Physics

−Hi+2,j+Hi,j+2. (6.8.20)

From (6.8.15),

v=h 00 −constant.

The derivatives ofvcan now be found in terms of thehijandHijwith the

aid of (6.8.20):

vx=H 10 h

2

00 ,

vxx=H 20 +H 11 − 3 h 00 H 10 +2h

3

00 ,

vxxx=12h

2

00 H^10 −^3 H

2

10 −^4 h^00 H^20 −^3 h^00 H^11 +3H^21

+H 30 − 2 h 10 h 01 − 6 h

4

00 ,

vy=h 00

̄

H 10 −

̄

H 20

vyy=2(h 10 h 20 +h 01 h 02 )−(h 10 h 02 +h 01 h 20 )

−h 00 (h

2

10 −h^10 h^01 +h

2

01 )+2h

2

00 h^11

− 2 h 00 H 21 +H 22 +h 00 H 30 −H 40 ,

vt=4(h 00 H 20 −h 10 h 01 −H 30 ). (6.8.21)

Hence,

vt+6v

2

x

+vxxx=3(h

2

10

+h

2

01

−h 00 H 11 +H 21 −H 30 ). (6.8.22)

The theorem appears after differentiating once again with respect tox.

6.8.2 The Wronskian Solution

The substitution

u=2D

2

x

(logw)

into the KP equation yields

(ut+6uux+uxxx)x+3uyy=2D

2

x

(

G

w

2

)

, (6.8.23)

where

G=wwxt−wxwt+3w

2

xx−^4 wxwxxx+wwxxxx+3(wwyy−w

2

y).

Hence, the KP equation will be satisfied if

G=0. (6.8.24)

The functionGis identical in form with the functionF in the first line

of (6.7.60) in the section on the KdV equation, but the symbolyin this

section and the symbolzin the KdV section have different origins. In this

section,yis one of the three independent variablesx,y, andtin the KP

equation whereasxandtare the only independent variables in the KdV

section andzis introduced to facilitate the analysis.