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182 4. The method of multiple scales

with the given form of solution

which requires

with (and still arbitrary. (We note that and that, with
we have so the classical result for group speed.)
The final stage is to find the solution—or, at least, the relevant part of the solution—
of equation (4.77c). The usual aim in such calculations is to find, completely, the first
term in (we hope) a uniformly valid asymptotic expansion. In this case we have yet
to find (although we know both c (k) and the determination of
arises from the terms in equation (4.77c). We will find just this one term; the rest
of the solution for is left as an exercise, the essential requirement being to check
that there are no inconsistencies that appear as is determined. These terms, in
(4.77c) give the equation

where the over-bar denotes the complex conjugate. When the earlier results are in-
corporated here, we find the equation for

This equation, (4.79), is a Nonlinear Schrödinger (NLS) equation, another of the ex-
tremely important exactly-integrable equations within the framework of soliton theory;
see Drazin & Johnson (1992). Thus we have a complete description of the first term
in the asymptotic expansion (4.76a,b):

where is a solution of (4.78) and both c (k) and are known.

We should comment that, because of the particular form of solution that we have
constructed in this example, it is appropriate only for certain types of initial data.
Thus, from (4.80), we see that, at t = 0, we must have an initial wave-profile that is
predominantly a harmonic wave, but one that admits a slow amplitude modulation i.e.