Page 254 - Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 239

Then,
y n+1 − y n−1 − (n +1)r 1 (y n )=0,
that is,

y (y n+1 − y n−1 )= n +1.

n
This equation will be referred to as the Milne-Thomson equation. Its origin
is distinct from that of the Toda equations, but it is of a similar nature and
clearly belongs to this section.

6.2.4 The Matsukidaira–Satsuma Equations
The following pairs of coupled differential–difference equations appeared in
a paper on nonlinear lattice theory published by Matsukidaira and Satsuma
in 1990.
The first pair is
q = q r (u r+1 − u r ),

r
u q
= .
r r
u r − u r−1 q r − q r−1
These equations contain two dependent variables q and u, and two indepen-
dent variables, x which is continuous and r which is discrete. The solution
is expressed in terms of a Hankel–Wronskian of arbitrary order n whose
elements are functions of x and r.
The second pair is

(q rs ) y = q rs (u r+1,s − u rs ),
q rs (v r+1,s − v rs )
= .
(u rs ) x
u rs − u r,s−1 q rs − q r,s−1
These equations contain three dependent variables, q, u, and v, and four
independent variables, x and y which are continuous and r and s which
are discrete. The solution is expressed in terms of a two-way Wronskian of
arbitrary order n whose elements are functions of x, y, r, and s.
In contrast with Toda equations, the discrete variables do not appear in
the solutions as orders of determinants.

6.2.5 The Korteweg–de Vries Equation
The Korteweg–de Vries (KdV) equation, namely
u t +6uu x + u xxx =0,

where the suffixes denote partial derivatives, is nonlinear and first arose in
1895 in a study of waves in shallow water. However, in the 1960s, interest in
the equation was stimulated by the discovery that it also arose in studies