  In this paper we study dispersive wave equation using the method of multiple
scales (MMS) and perform several numerical tests to investigate its accuracy.
The key feature of our MMS solution is the linearity of the amplitude equation
and the complex nature of the time-frequency. The MMS is tested as an initial
value problem using three choices of the dispersion model, one toy and two
Lorentz models. Depending on the parameters of the problem, the amplitude
equation can be both well- or ill-posed. Despite the ill-posedness, the MMS
solution remains a valid approximation of the solution to the original
nonlinear model.
