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where In this problem we seek a solution which depends on x – c t (here,
c is the phase speed), on is the group speed) and on t (time). We expect
that the dependence on is slow i.e. in the form but we will allow
an even slower time dependence; we define

although we have yet to determine c and With and
(4.74), equation (4.73) becomes

and we seek a solution

with ‘cc’ denotes the complex conjugate. This solution represents a primary
harmonic wave together with appropriate higher harmonics (the number
of which depends on i.e. on the hierarchy of nonlinear interactions). The wave
number of the primary wave, k, is given and real; X and T are real (of course), but
each is complex-valued. We will assume that c and are independent of
First we use (4.76a) in (4.75) to give the sequence of equations

as far as the terms. The solution of (4.77a) is given in the form

(see (4.76b)), and so we-require

which defines the phase speed; at this stage is unknown. Equation (4.77b), for
now becomes