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

LetA

(n)

(r, s) denote the two-way Wronskian of orderndefined as follows:

A

(n)

(r, s)=

∣

∣

fr+i− 1 ,s+j− 1

∣

∣

n

, (6.6.7)

wherefrs=frs(x, y), (frs)x=fr,s+1, and (frs)y=fr+1,s.

Let

τrs=A

(n)

(r, s). (6.6.8)

Theorem 6.10.
∣
∣
∣
∣

τr+1,s τr+1,s− 1

τrs τr,s− 1

∣

∣

∣

∣

∣

∣

∣

∣

(τrs)xy (τrs)y

(τrs)x τrs

∣

∣

∣

∣

=

∣

∣

∣

∣

(τrs)y (τr,s− 1 )y

τrs τr,s− 1

∣

∣

∣

∣

∣

∣

∣

∣

(τr+1,s)x τr+1,s

(τrs)x τrs

∣

∣

∣

∣

for all values ofnand all differentiable functionsfrs(x, y).

Proof.

τrs=A

(n+1)

n+1,n+1

(r, s),

τr+1,s=−A

(n+1)

1 ,n+1

(r, s),

τr,s+1=−A

(n+1)

n+1, 1 (r, s),

τr+1,s+1=A

(n+1)

11

(r, s).

Hence, applying the Jacobi identity,

∣

∣

∣

∣

τr+1,s+1 τr+1,s+1

τr,s+1 τrs

∣

∣

∣

∣

=A

(n+1)

(r, s)A

(n+1)

1 ,n+1;1,n+1

(r, s)

=A

(n+1)

(r, s)A

(n−1)

(r+1,s+1).

Replacingsbys−1,

∣

∣

∣

∣

τr+1,s τr+1,s− 1

τrs τr,s− 1

∣

∣

∣

∣

=A

(n+1)

(r, s−1)A

(n−1)

(r+1,s), (6.6.9)

(τrs)x=−A

(n+1)

n+1,n

(r, s),

(τrs)y=−A

(n+1)

n,n+1(r, s),

(τrs)xy=A

(n+1)

nn

(r, s).

Hence, applying the Jacobi identity,

∣

∣

∣

∣

(τrs)xy (τrs)y

(τrs)x τrs

∣

∣

∣

∣

=A

(n+1)

(r, s)A

(n+1)

n,n+1;n,n+1

(r, s)

=A

(n+1)

(r, s)A

(n−1)

(r, s) (6.6.10)

(τr,s+1)y=−A

(n+1)

n 1

(r, s).