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Thus for a periodic solution to exist, we require the initial amplitude, a, to be
restricted in value, a conclusion that can also be reached on the basis of an examina-
tion of the direction field for this equation. For a given the amplitude
is chosen by selecting and which satisfy (5.54) and (5.55),
where is given by (5.56). The problem of the existence of solutions to this set—
solutions do exist!—is left as an additional investigation.

This example has demonstrated how we can extract fairly simple estimates from a
solution with a complicated structure, even though the governing differential equation
may have persuaded us that no serious difficulties would be encountered.
Finally, we apply the method of multiple scales to a partial differential equation of
some importance.

E5.7 Klein-Gordon equation
The general form of the Klein-Gordon equation, written in one spatial dimension, is

where V(u) (which can be taken as a potential, in quantum-mechanical terms) is,
typically, a function with nonlinearity more severe than quadratic. We will consider
the problem for which

and so introduce a parameter which we will allow to satisfy The equa-
tion with this choice arises, for example, in the study of wave propagation in a cold
plasma. (The choice V(u) = –cos u gives rise to the so-called sine-Gordon equation—a
pun on the original name—and this equation has exact ‘soliton’ solutions; see e.g.
Drazin & Johnson, 1992.) We follow the technique described in §4.4, and so we intro-
duce

and then with we obtain the equation

We seek a solution which is to be periodic in in the form