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118 3. Further applications

and so we have

This two-term asymptotic expansion is clearly uniformly valid for and for
and it is also analytic in this domain, so the use of the Taylor expansion to generate
(3.6) is justified. It is left as an exercise (Q3.1) to find the next term in this asymp-
totic expansion and then to discuss further the validity of the expansion; it is indeed
uniformly valid. A few other examples of regular expansions obtained from problems
posed using partial differential equations can be found in Q3.2 and Q3.3.
This exercise has demonstrated that, as with analogous problems based on ordinary
differential equations, we may encounter asymptotic expansions that are essentially
uniformly valid i.e. the problem is regular. However, this is very much a rarity: most
problems that we meet, and that are of interest, turn out to be singular perturbation
problems. We now discuss this aspect, in the context of partial differential equations,
and highlight the two main types of non-uniformity that can arise.

3.2 SINGULAR PROBLEMS I
The most straightforward type of non-uniformity, as we have seen for ordinary differ-
ential equations, arises when the asymptotic expansion that has been obtained breaks
down and thereby leads to the introduction of a new, scaled variable. This situation is
typical of some wave propagation problems, for which an asymptotic expansion valid
near the initial data becomes non-uniform for later times/large distances. Indeed, the
general structure of such problems is readily characterised by an expansion of the form

where c is the speed of the wave and the functions f ,g and h are bounded (and typically
well-behaved, often decaying for However, for a solution defined
in (which is expected in wave problems), we clearly have a breakdown when
irrespective of the value (size) of (x – c t). Thus we would need to examine
the problem in the far-field, defined by the new variables (In this
example, we have used x – ct (c > 0) for right-running waves; correspondingly, for
left-running waves, we would work with x + ct.) We present an example of this type
of singular perturbation problem.

E3.1 Nonlinear, dispersive wave propagation
A model equation which describes small-amplitude, weakly dispersive water waves can
be written as