Page 318 - Determinants and Their Applications in Mathematical Physics

6.11 The Relativistic Toda Equation — A Brief Note 303

+ ct
x = t 1 − a = √ . (6.11.4)
2
1+ c 2
Equations (6.11.1)–(6.11.3) are satisfied by the functions
U n = |u i,n+j−1 | m ,
V n = |v i,n+j−1 | m ,
W n = |w i,n+j−1 | m , (6.11.5)
where the determinants are Casoratians (Section 4.14) of arbitrary order
m whose elements are given by
u ij = F ij + G ij ,
1
v ij = a i F ij + G ij ,
1 a i
w ij = F ij + a i G ij , (6.11.6)
where a i

1

j
F ij = exp(ξ i ),
a i − a

j
G ij = a i exp(η i ),
x 1 − aa i
ξ i = + b i ,
a i
η i = a i x + c i , (6.11.7)
and where the a i , b i , and c i are arbitrary constants.