Page 276 - Determinants and Their Applications in Mathematical Physics

6.6 The Matsukidaira–Satsuma Equations 261

6.6.2 A System With Two Continuous and Two Discrete
Variables

Let A (n) (r, s) denote the two-way Wronskian of order n defined as follows:

A (n) (r, s)= f r+i−1,s+j−1 , (6.6.7)

n
where f rs = f rs (x, y), (f rs ) x = f r,s+1 , and (f rs ) y = f r+1,s .
Let
τ rs = A (n) (r, s). (6.6.8)
Theorem 6.10.

τ

r+1,s τ r+1,s−1 (τ rs ) xy (τ rs ) y
τ r,s−1 (τ rs ) x τ rs

τ rs

= (τ rs ) y (τ r,s−1 ) y (τ r+1,s ) x τ r+1,s
τ r,s−1 (τ rs ) x τ rs

τ rs
for all values of n and all differentiable functions f rs (x, y).
Proof.
(n+1)
τ rs = A (r, s),
n+1,n+1
(n+1)
τ r+1,s = −A 1,n+1 (r, s),
(n+1)
τ r,s+1 = −A (r, s),
n+1,1
(n+1)
τ r+1,s+1 = A (r, s).
11
Hence, applying the Jacobi identity,

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

τ r,s+1 τ rs
1,n+1;1,n+1
= A (n+1) (r, s)A (n−1) (r +1,s +1).
Replacing s by s − 1,

τ (n+1) (n−1)
r+1,s τ r+1,s−1 = A (r, s − 1)A (r +1,s), (6.6.9)
τ r,s−1

τ rs
(n+1)
(τ rs ) x = −A (r, s),
n+1,n
(n+1)
(τ rs ) y = −A (r, s),
n,n+1
(τ rs ) xy = A (n+1) (r, s).
nn
Hence, applying the Jacobi identity,

(n+1) (n+1)
(τ rs ) xy (τ rs ) y = A (r, s)A n,n+1;n,n+1 (r, s)
(τ rs ) x τ rs
= A (n+1) (r, s)A (n−1) (r, s) (6.6.10)
(n+1)
(τ r,s+1 ) y = −A (r, s).
n1