Page 280 - Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 265

Eliminating the sum common to (6.7.3) and (6.7.5) and the sum common
to (6.7.4) and (6.7.6),

v = D x (log A)= A rs − b r , (6.7.7)
r s r

1
ij
D x (A )= (b i + b j )A − A A . (6.7.8)
is
ij
rj
2
r s
Returning to (A) and (B),
3
D t (log A)= b e r A , (6.7.9)
rr
r
r
3
D t (A )= − b e r A A . (6.7.10)
rj
ir
ij
r
r
3
Now return to (C) and (D) with f r = g r = b .
r
3 2 2 3
b e r A rr + (b − b r b s + b )A rs = b , (6.7.11)
r r s r
r r s r
3 2 2
b e r A A rj + (b − b r b s + b )A A rj
is
ir
r r s
r r s
3
3
1
= (b + b )A . (6.7.12)
ij
2 i j
Eliminating the sum common to (6.7.9) and (6.7.11) and the sum common
to (6.7.10) and (6.7.12),
3 2 2
D t (log A)= b − (b − b r b s + b )A , (6.7.13)
rs
r r s
r r s
2 2 1 3 3
D t (A )= (b − b r b s + b )A A rj − (b + b )A .(6.7.14)
is
ij
ij
2
r s i j
r s
The derivatives v x and v t can be evaluated in a convenient form with the
aid of two functions ψ is and φ ij which are defined as follows:

ψ is = b A , (6.7.15)
rs
i
r
r

φ ij = b ψ is ,
j
s
s

= b b A rs
i j
r s
r s
= φ ji . (6.7.16)
They are definitions of ψ is and φ ij .
Lemma. The function φ ij satisfies the three nonlinear recurrence rela-
tions:
1
a. φ i0 φ j1 − φ j0 φ i1 = (φ i+2,j − φ i,j+2 ),
2
1
b. D x (φ ij )= (φ i+1,j + φ i,j+1 ) − φ i0 φ j0 ,
2