Page 275 - Determinants and Their Applications in Mathematical Physics

260 6. Applications of Determinants in Mathematical Physics

Applying the Jacobi identity,

F 11
FF 13,13 = F 13

F 31 F 33
=0.
But F 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 = q r (u r+1 − u r ),

r
u q
b. r = r ,
u r − u r−1 q r − q r−1
where q r and u r are functions of x, are satisfied by the functions
τ r+1
q r = τ r ,
τ
u r = τ r r

for all values of n and all differentiable functions f s (x).
Proof.

F 31

q = − ,
τ
r 2
r
F 33
q r − q r−1 = − τ r−1 τ r ,
F 11

u = ,
τ
r 2
r
F 13
u r − u r−1 = τ r−1 τ r ,
F 31
u r+1 − u r = − .
τ r τ r+1
Hence,
q τ r+1
= τ r = q r ,
r
u r+1 − u r
which proves (a) and
u (q r − q r−1 ) F 11 F 33

=
r
q (u r − u r−1 ) F 31 F 13

r
=1,
which proves (b).