262 6. Applications of Determinants in Mathematical Physics
Hence,
∣
∣
∣
∣
(τr,s+1)y (τrs)y
τr,s+1 τrs
∣
∣
∣
∣
=A
(n+1)
(r, s)A
(n+1)
n,n+1;1,n+1
(r, s)
=A
(n+1)
(r, s)A
(n)
n 1
(r, s)
=A
(n+1)
(r, s)A
(n−1)
(r, s+1).
Replacingsbys−1,
∣
∣
∣
∣
(τrs)y (τr,s− 1 )y
τrs τr,s− 1
∣
∣
∣
∣
=A
(n+1)
(r, s−1)A
(n−1)
(r, s), (6.6.11)
(τr+1,s)x=A
(n+1)
1 n
(r, s).
Hence,
∣
∣
∣
∣
(τr+1,s)x τr+1,s
(τrs)x τrs
∣
∣
∣
∣
=A
(n+1)
(r, s)A
(n+1)
1 ,n+1;n,n+1
(r, s)
=A
(n+1)
(r, s)A
(n−1)
(r+1,s). (6.6.12)
Theorem 6.10 follows from (6.6.9)–(6.6.12).
Theorem 6.11.
∣ ∣ ∣ ∣ ∣ ∣
τr+1,s− 1 τr,s− 1 (τr,s− 1 )y
τr+1,s τrs (τrs)y
(τr+1,s)x (τrs)x (τrs)xy
∣ ∣ ∣ ∣ ∣ ∣
=0.
Proof. Denote the determinant byG. Then, Theorem 6.10 can be
expressed in the form
G 33 G 11 =G 31 G 13. (6.6.13)
Applying the Jacobi identity,
GG 13 , 13 =
∣
∣
∣
∣
G 11 G 13
G 31 G 33
∣
∣
∣
∣
=0.
ButG 13 , 13 = 0. The theorem follows.
Theorem 6.12. The Matsukidaira–Satsuma equations with two contin-
uous independent variables, two discrete independent variables, and three
dependent variables, namely
a.(qrs)y=qrs(ur+1,s−urs),
b.
(urs)x
urs−ur,s− 1
=
(vr+1,s−vrs)qrs
qrs−qr,s− 1
,
whereqrs,urs, and vrsare functions of xandy, are satisfied by the
functions
qrs=
τr+1,s
τrs