Page 295 - Determinants and Their Applications in Mathematical Physics

280 6. Applications of Determinants in Mathematical Physics

−H i+2,j + H i,j+2 . (6.8.20)
From (6.8.15),
v = h 00 − constant.

The derivatives of v can now be found in terms of the h ij and H ij with the
aid of (6.8.20):
2
v x = H 10 h ,
00
3
v xx = H 20 + H 11 − 3h 00 H 10 +2h ,
00
2
v xxx =12h H 10 − 3H 2
00 10 − 4h 00 H 20 − 3h 00 H 11 +3H 21
4
+H 30 − 2h 10 h 01 − 6h ,
00
¯ ¯
v y = h 00 H 10 − H 20
v yy =2(h 10 h 20 + h 01 h 02 ) − (h 10 h 02 + h 01 h 20 )
−h 00 (h 2 2 2
00
10 − h 10 h 01 + h )+2h h 11
01
−2h 00 H 21 + H 22 + h 00 H 30 − H 40 ,
v t =4(h 00 H 20 − h 10 h 01 − H 30 ). (6.8.21)
Hence,
2 2 2
v t +6v + v xxx =3(h 10 + h 01 − h 00 H 11 + H 21 − H 30 ). (6.8.22)
x
The theorem appears after differentiating once again with respect to x.
6.8.2 The Wronskian Solution

The substitution
2
u =2D (log w)
x
into the KP equation yields

G
2
(u t +6uu x + u xxx ) x +3u yy =2D , (6.8.23)
w
x 2
where
2
G = ww xt − w x w t +3w 2 − 4w x w xxx + ww xxxx +3(ww yy − w ).
xx y
Hence, the KP equation will be satisfied if
G =0. (6.8.24)
The function G is identical in form with the function F in the first line
of (6.7.60) in the section on the KdV equation, but the symbol y in this
section and the symbol z in the KdV section have different origins. In this
section, y is one of the three independent variables x, y, and t in the KP
equation whereas x and t are the only independent variables in the KdV
section and z is introduced to facilitate the analysis.