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176 4. The method of multiple scales

where we have used (4.55) (and note that thus matching occurs if we
choose

For Z > 0, X > 0, from (4.57) and (4.54a), we have

from (4.56a) we obtain

which also matches if we choose

The matching conditions are usually called, in this context, connection formulae: they
‘connect’ the solutions on either side of the turning point i.e. the relation between
and Here, we have three relations between the four constants and
so only one is free; that only one occurs here is because, of the two (independent)
boundary conditions that we may prescribe, one has been fixed by seeking a bounded
solution in X < 0 i.e. the exponentially growing solution has already been excluded.

A number of other examples of turning-point problems are offered in the exercises;
see Q4.18–4.21. This completes all that we will write about the routine applications to
ordinary differential equations; we now take a brief look at how these same techniques
are relevant to the study of partial differential equations.

4.4 APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS
It is no accident that we will discuss partial differential equations which are associated
with wave propagation; this type of equation is analogous to oscillatory solutions of
ordinary differential equations. (These two categories of equations are the most natural
vehicles for the method of multiple scales, although others are certainly possible.) In
particular we will start with an equation that has become a classical example of its type:
Bretherton’s model equation for the weak, nonlinear interaction of dispersive waves
(Bretherton, 1964).