Determinants and Their Applications in Mathematical Physics

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

,