Determinants and Their Applications in Mathematical Physics

238 6. Applications of Determinants in Mathematical Physics

The same substitutions transform the second-order equation first into

D

2

(logyn)=yn+1− 2 yn+yn− 1

and then into

D

2

(logun)=

un+1un− 1

u

2

n

. (6.2.3)

Other equations which are similar in nature to the transformed second-

order Toda equations are

DxDy(logun)=

un+1un− 1

u

2

n

,

(D

2

x

+D

2

y

) logun=

un+1un− 1

u

2

n

,

1

ρ

Dρ

[

ρDρ(logun)

]

=

un+1un− 1

u^2

n

. (6.2.4)

All these equations are solved in Section 6.5.

Note that (6.2.1) can be expressed in the form

D(yn)=yn(yn+1−yn− 1 ), (6. 2 .1a)

which appeared in 1974 in a paper by Zacharov, Musher, and Rubenchick

on Langmuirwaves in aplasma and was solved in 1987 by S. Yamazaki

in terms of determinantsP 2 n− 1 andP 2 nof ordern. Yamazaki’s analysis

involves a continued fraction. The transformed equation (6.2.2) is solved

below without introducing a continued fraction but with the aid of the

Jacobi identity and one of its variants (Section 3.6).

The equation

DxDy(Rn) = exp(Rn+1−Rn)−exp(Rn−Rn− 1 ) (6.2.5)

appears in a 1991 paper by Kajiwara and Satsuma on theq-difference

version of the second-order Toda equation.

The substitution

Rn= log

(

un+1

un

)

reduces it to the first line of (6.2.4).

In the chapter on reciprocal differences in his bookCalculus of Finite

Differences, Milne-Thomson defines an operatorrnby the relations

r 0 f(x)=f(x),

r 1 f(x)=

1

f′(x)

,

[

rn+1−rn− 1 −(n+1)r 1 rn

]

f(x)=0.

Put

rnf=yn.