Multiple scale fourier transformation: An application to nonlinear dispersive waves

A Fourier transformation involving multiple scales is applied to describe the far-field asymptotic behaviour of nonlinear dispersive waves. It is shown that a nonlinear asymptotic perturbation can be carried out in terms of simple calculations with respect to Dirac delta functions involving a multiple scale wave number and frequency space. Fourier transformed versions of the nonlinear Schrödinger and Korteweg-de Vries equations are derived explicitly.