Page 253 - 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
2
D (log y n )= y n+1 − 2y n + y n−1
and then into
2
D (log u n )= u n+1 u n−1 . (6.2.3)
u 2
n
Other equations which are similar in nature to the transformed second-
order Toda equations are
u n+1 u n−1
D x D y (log u n )= ,
u 2
n
2
2
(D + D ) log u n = u n+1 u n−1 ,
u
x y 2
n
1 u n+1 u n−1
D ρ ρD ρ (log u n ) = . (6.2.4)
ρ u 2
n
All these equations are solved in Section 6.5.
Note that (6.2.1) can be expressed in the form
D(y n )= y n (y n+1 − y n−1 ), (6.2.1a)
which appeared in 1974 in a paper by Zacharov, Musher, and Rubenchick
on Langmuir waves in a plasma and was solved in 1987 by S. Yamazaki
in terms of determinants P 2n−1 and P 2n of order n. 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
D x D y (R n ) = exp(R n+1 − R n ) − exp(R n − R n−1 ) (6.2.5)
appears in a 1991 paper by Kajiwara and Satsuma on the q-difference
version of the second-order Toda equation.
The substitution

u n+1
R n = log u n
reduces it to the first line of (6.2.4).
In the chapter on reciprocal differences in his book Calculus of Finite
Differences, Milne-Thomson defines an operator r n by the relations
r 0 f(x)= f(x),
1
r 1 f(x)= ,
f (x)

r n+1 − r n−1 − (n +1)r 1 r n f(x)=0.
Put
r n f = y n .