Page 294 - Determinants and Their Applications in Mathematical Physics

6.8 The Kadomtsev–Petviashvili Equation 279

new formulae for the derivatives of log A:

v = D x (log A)= A rs − λ r , (6.8.15)
r,s r

D y (log A)= − (b r − c s )A rs + λ r µ r , (6.8.16)
r,s r
2 2
D t (log A)= −4 (b − b r c s + c )A rs
r s
r,s
2 2
+4 λ r (b − b r c r + c ). (6.8.17)
r r
r
Equations (6.8.16) and (6.8.17) are not applied below but have been
included for their interest.
Eliminating the sum common to (6.8.6) and (6.8.12), the sum common
to (6.8.7) and (6.8.13), and the sum common to (6.8.8) and (6.8.14), we
find new formulas for the derivatives of A :
ij

is
rj
ij
ij
D x (A )=(b i + c j )A − A A ,
r,s
2
2
D y (A )= −(b − c )A + (b r − c s )A A ,
is
ij
ij
rj
i j
r,s
2 2
3
3
D t (A )= −4(b + c )A +4 (b − b r c s + c )A A . (6.8.18)
rj
is
ij
ij
i j r s
r,s
Define functions h ij , H ij , and H ij as follows:
n n

h ij = b c A ,
i j
rs
r s
r=1 s=1
H ij = h ij + h ji = H ji ,
H ij = h ij − h ji = −H ji . (6.8.19)
The derivatives of these functions are found by applying (6.8.18):

D x (h ij )= b c (b r + c s )A rs − A A ps
i j
rq
r s
r,s p,q

= b c (b r + c s )A rs − b A rq c A ps
j
i j
i
r s r s
r,s r,q p,s
= h i+1,j + h i,j+1 − h i0 h 0j ,
which is a nonlinear differential recurrence relation. Similarly,
D y (h ij )= h i0 h 1j − h i1 h 0j − h i+2,j + h i,j+2 ,
D t (h ij )=4(h i0 h 2j − h i1 h 1j + h i2 h 0j − h i+3,j − h i,j+3 ),
D x (H ij )= H i+1,j + H i,j+1 − h i0 h 0j − h 0i h j0 ,
D y (H ij )=(h i0 h 1j + h 0i h j1 ) − (h i1 h 0j + h 1i h j0 )