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218 5. Some worked examples arising from physical problems

and so we obtain the set of equations

where we have elected to take k (and hence as constants, for the purposes of this
discussion. A suitable solution of equation (5.58) is

with and are arbitrary functions. Equation (5.59), for now
becomes

where We use and then observe that is periodic in
i.e. in only if

which have the general solutions

for arbitrary functions F and Thus the leading term in the asymptotic expansion
is

where and we observe that we may write the oscillatory
term as

Both the wave number (k) and the frequency have small-amplitude corrections—
we assume that is a bounded function—which produces an approximate speed
of the carrier wave of

which shows that the inclusion of the nonlinearity in the equation is to change the