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=Dx(logA)=

∑

r

∑

s

A

rs

−

∑

r

br, (6.7.7)

Dx(A

ij

)=

1

2

(bi+bj)A

ij

−

∑

r

∑

s

A

is

A

rj

. (6.7.8)

Returning to (A) and (B),

Dt(logA)=

∑

r

b

3

rerA

rr

, (6.7.9)

Dt(A

ij

)=−

∑

r

b

3

r

erA

ir

A

rj

. (6.7.10)

Now return to (C) and (D) withfr=gr=b
3
r

.

∑

r

b

3

r

erA

rr

+

∑

r

∑

s

(b

2

r

−brbs+b

2

s

)A

rs

=

∑

r

b

3

r

, (6.7.11)

∑

r

b

3

r

erA

ir

A

rj

+

∑

r

∑

s

(b

2

r

−brbs+b

2

s

)A

is

A

rj

=

1

2

(b

3

i+b

3

j)A

ij

. (6.7.12)

Eliminating the sum common to (6.7.9) and (6.7.11) and the sum common

to (6.7.10) and (6.7.12),

Dt(logA)=

∑

r

b

3

r−

∑

r

∑

s

(b

2

r−brbs+b

2

s)A

rs

, (6.7.13)

Dt(A

ij

)=

∑

r

∑

s

(b

2

r−brbs+b

2

s)A

is

A

rj

−

1

2

(b

3

i+b

3

j)A

ij

.(6.7.14)

The derivativesvxandvtcan be evaluated in a convenient form with the

aid of two functionsψisandφijwhich are defined as follows:

ψis=

∑

r

b

i

rA

rs

, (6.7.15)

φij=

∑

s

b

j

sψis,

=

∑

r

∑

s

b

i

rb

j

sA

rs

=φji. (6.7.16)

They are definitions ofψisandφij.

Lemma.The function φij satisfies the three nonlinear recurrence rela-

tions:

a.φi 0 φj 1 −φj 0 φi 1 =

1

2

(φi+2,j−φi,j+2),

b.Dx(φij)=

1

2

(φi+1,j+φi,j+1)−φi 0 φj 0 ,