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180 4. The method of multiple scales

Equation (4.70), after multiplication by can be written as

or

which describes the property (the wave action; see E4.5) propagating at the group
speed, and evolving by virtue of the non-zero right-hand side of this equation.
Similarly, equation (4.69) can be written first as

and if we elect to write then (4.61) and (4.64) allow us to interpret
and and then we obtain

Thus the correction to the phase, also propagates at the group speed and evolves.
Finally, from (4.63), and noting that (4.68) (or (4.72)) implies we obtain

the wave number (and, correspondingly, the frequency) propagate unchanged at the
group speed. (The initial data will include the specification of

This example has demonstrated that the method of multiple scales can be used to
analyse appropriate partial differential equations, even if we had to tease out the details
by introducing, in particular, the group speed. A related exercise can be found in
Q4.22.
In our analysis of Bretherton’s equation, we worked—not surprisingly—with real-
valued functions throughout i.e. There is, sometimes, an advantage in working
within a complex-valued framework e.g. complex conjugate. We will use this
approach in the next example, but also take note of the general structure that is evident
in E4.7.

E4.8 A nonlinear wave equation: the NLS equation
A wave profile, satisfies the equation