Determinants and Their Applications in Mathematical Physics

260 6. Applications of Determinants in Mathematical Physics

Applying the Jacobi identity,

FF 13 , 13 =

∣

∣

∣

∣

F 11 F 13

F 31 F 33

∣

∣

∣

∣

=0.

ButF 13 , 13 = 0. The theorem follows.

Theorem 6.9. The Matsukidaira–Satsuma equations with one continuous

independent variable, one discrete independent variable, and two dependent

variables, namely

a.q

′

r

=qr(ur+1−ur),

b.

u

′

r

ur−ur− 1

=

q

′

r

qr−qr− 1

,

whereqrandurare functions ofx, are satisfied by the functions

qr=

τr+1

τr

,

ur=

τ

′

r

τr

for all values ofnand all differentiable functionsfs(x).

Proof.

q

′

r=−

F 31

τ

2

r

,

qr−qr− 1 =−

F 33

τr− 1 τr

,

u

′

r=

F 11

τ

2

r

,

ur−ur− 1 =

F 13

τr− 1 τr

,

ur+1−ur=−

F 31

τrτr+1

.

Hence,

q

′

r

ur+1−ur

=

τr+1

τr

=qr,

which proves (a) and

u

′

r

(qr−qr− 1 )

q

′

r(ur−ur−^1 )

=

F 11 F 33

F 31 F 13

=1,

which proves (b).