Determinants and Their Applications in Mathematical Physics

6.5 The Toda Equations 255

The derivative ofAnwith respect tox, as obtained by differentiating the

rows, consists of the sum ofndeterminants, only one of which is nonzero.

That determinant is a cofactor ofAn+1:

Dx(An)=−A

(n+1)

n,n+1

.

Differentiating the columns with respect toyand then the rows with respect

tox,

Dy(An)=−A

(n+1)

n+1,n,

DxDy(An)=A

(n+1)

nn

. (6.5.7)

Denote the determinant in (6.5.6) byE. Then, applying the Jacobi identity

(Section 3.6) toAn+1,

E=

∣

∣

∣

∣

∣

A

(n+1)

nn −A

(n+1)

n,n+1

−A

(n+1)

n+1,n A

(n+1)

n+1,n+1

∣

∣

∣

∣

∣

=An+1A

(n+1)

n,n+1;n,n+1

which simplifies to the right side of (6.5.6).

It follows as a corollary that the equation

D

2

(logun)=

un+1un− 1

u^2

n

,D=

d

dx

,

is satisfied by the Hankel–Wronskian

un=An=|D

i+j− 2

(f)|n,

where the functionf=f(x) is arbitrary.

Theorem 6.4. The equation

1

ρ

Dρ

[

ρDρ(logun)

]

=

un+1un− 1

u

2

n

,Dρ=

d

dρ

,

is satisfied by the function

un=An=e

−n(n−1)x

Bn, (6.5.8)

where

Bn=

∣

∣

(ρDρ)

i+j− 2

f(ρ)

∣

∣

n

,f(ρ)arbitrary.

Proof. Putρ=e
x

. Then,ρDρ=Dxand the equation becomes

D

2

x(logAn)=

ρ

2

An+1An− 1

A^2

n

. (6.5.9)

Applying (6.5.8) to transform this equation fromAntoBn,

D

2

x(logBn)=D

2

x(logAn)

=

ρ

2

Bn+1Bn− 1

B

2

n

e

−[(n+1)n+(n−1)(n−2)− 2 n(n−1)]x