Page 290 - Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 275

2
Hence, ww zz − w =0,
z
2
F = ww xt − w x w t +3w 2 − 4w x w xxx + ww xxxx +3(ww zz − w )
xx z

= w (w t +4w xxx ) x − 3(w xxxx − w zz )
2
−w x (w t +4w xxx )+3(w 2 − w ). (6.7.60)
xx z
The evaluation of the derivatives of a Wronskian is facilitated by expressing
it in column vector notation.
Let
..............................
W = C 0 C 1 ··· C n−4 C n−3 C n−2 C n−1 , (6.7.61)

n
where
T
C j = D (ψ 1 ) D (ψ 2 ) ··· D (ψ n ) .
j
j
j
x x x
The significance of the row of dots above the (n − 3) columns C 0 to C n−4
will emerge shortly. It follows from (6.7.58) and (6.7.59) that
D x (C j )= C j+1 ,
2
D z (C j )= D (C j )= C j+2 ,
x
3
D t (C j )= −4D (C j )= −4C j+3 . (6.7.62)
x
Hence, differentiating (6.7.61) and discarding determinants with two
identical columns,
..............................
w x = C 0 C 1 ··· C n−4 C n−3 C n−2 C n ,

n
..............................

w xx = C 0 C 1 ··· C n−4 C n−3 C n−1 C n
n
..............................
+ C 0 C 1 ··· C n−4 C n−3 C n−2 C n+1 ,

n
..............................
w z = C 0 C 1 ··· C n−4 C n−3 C n C n−1

n
..............................
+ C 0 C 1 ··· C n−4 C n−3 C n−2 C n+1 ,

n
etc. The significance of the row of dots above columns C 0 to C n−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
..............................
V pqr = C 0 C 1 ··· C n−4 C p C q C r . (6.7.63)

n
Then, V pqr =0 if p, q, and r are not distinct and V qpr = −V pqr , etc. In this
notation,
w = V n−3,n−2,n−1 ,
w x = V n−3,n−2,n ,
w xx = V n−3,n−1,n + V n−3,n−2,n+1 ,
w xxx = V n−2,n−1,n +2V n−3,n−1,n+1 + V n−3,n−2,n+2 ,