Determinants and Their Applications in Mathematical Physics

6.8 The Kadomtsev–Petviashvili Equation 279

new formulae for the derivatives of logA:

v=Dx(logA)=

∑

r,s

A

rs

−

∑

r

λr, (6.8.15)

Dy(logA)=−

∑

r,s

(br−cs)A

rs

+

∑

r

λrμr, (6.8.16)

Dt(logA)=− 4

∑

r,s

(b

2

r−brcs+c

2

s)A

rs

+4

∑

r

λr(b

2

r

−brcr+c

2

r

). (6.8.17)

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 ofA

ij

:

Dx(A

ij

)=(bi+cj)A

ij

−

∑

r,s

A

is

A

rj

,

Dy(A

ij

)=−(b

2

i

−c

2

j

)A

ij

+

∑

r,s

(br−cs)A

is

A

rj

,

Dt(A

ij

)=−4(b

3

i

+c

3

j

)A

ij

+4

∑

r,s

(b

2

r

−brcs+c

2

s

)A

is

A

rj

.(6.8.18)

Define functionshij,Hij, andHijas follows:

hij=

n

∑

r=1

n

∑

s=1

b

i

r

c

j

s

A

rs

,

Hij=hij+hji=Hji,

Hij=hij−hji=−Hji. (6.8.19)

The derivatives of these functions are found by applying (6.8.18):

Dx(hij)=

∑

r,s

b

i

rc

j

s

[

(br+cs)A

rs

−

∑

p,q

A

rq

A

ps

]

=

∑

r,s

b

i

rc

j

s(br+cs)A

rs

−

∑

r,q

b

i

rA

rq

∑

p,s

c

j

sA

ps

=hi+1,j+hi,j+1−hi 0 h 0 j,

which is a nonlinear differential recurrence relation. Similarly,

Dy(hij)=hi 0 h 1 j−hi 1 h 0 j−hi+2,j+hi,j+2,

Dt(hij)=4(hi 0 h 2 j−hi 1 h 1 j+hi 2 h 0 j−hi+3,j−hi,j+3),

Dx(Hij)=Hi+1,j+Hi,j+1−hi 0 h 0 j−h 0 ihj 0 ,

Dy(Hij)=(hi 0 h 1 j+h 0 ihj 1 )−(hi 1 h 0 j+h 1 ihj 0 )