Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 239

Then,

yn+1−yn− 1 −(n+1)r 1 (yn)=0,

that is,

y

′

n(yn+1−yn−^1 )=n+1.

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

′

r=qr(ur+1−ur),

u

′

r

ur−ur− 1

=

q

′

r

qr−qr− 1

These equations contain two dependent variablesqandu, and two indepen-

dent variables,xwhich is continuous andrwhich is discrete. The solution

is expressed in terms of a Hankel–Wronskian of arbitrary ordernwhose

elements are functions ofxandr.

The second pair is

(qrs)y=qrs(ur+1,s−urs),

(urs)x

urs−ur,s− 1

=

qrs(vr+1,s−vrs)

qrs−qr,s− 1

These equations contain three dependent variables,q,u, andv, and four

independent variables,xandywhich are continuous andrandswhich

are discrete. The solution is expressed in terms of a two-way Wronskian of

arbitrary ordernwhose elements are functions ofx,y,r, ands.

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

ut+6uux+uxxx=0,

where the suffixes denote partial derivatives, is nonlinear and first arose in

1895 in a study ofwaves inshallow water. However, in the 1960s, interest in

the equation was stimulated by the discovery that it also arose in studies