Page 277 - Determinants and Their Applications in Mathematical Physics

262 6. Applications of Determinants in Mathematical Physics

Hence,

(n+1) (n+1)
= A (r, s)A (r, s)
(τ r,s+1 ) y
n,n+1;1,n+1
(τ rs ) y
τ r,s+1 τ rs

= A (n+1) (r, s)A (n) (r, s)
n1
= A (n+1) (r, s)A (n−1) (r, s +1).
Replacing s by s − 1,

(n+1) (n−1)
(τ rs ) y (τ r,s−1 ) y = A (r, s − 1)A (r, s), (6.6.11)
τ r,s−1

τ rs
(n+1)
(τ r+1,s ) x = A (r, s).
1n
Hence,

(n+1) (n+1)
= A (r, s)A (r, s)
(τ r+1,s ) x
1,n+1;n,n+1
τ r+1,s

(τ rs ) x τ rs
= 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.

τ r+1,s−1 τ r,s−1

(τ r,s−1 ) y
=0.
τ r+1,s τ rs
(τ rs ) y
(τ r+1,s ) x (τ rs ) x (τ rs ) xy
Proof. Denote the determinant by G. 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,

G 11
GG 13,13 = G 13
G 31 G 33

=0.
But G 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. (q rs ) y = q rs (u r+1,s − u rs ),
b. (u rs ) x = (v r+1,s − v rs )q rs ,
u rs − u r,s−1 q rs − q r,s−1
where q rs , u rs , and v rs are functions of x and y, are satisfied by the
functions
q rs = τ r+1,s ,
τ rs