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 ,