Two-timing procedure for higher-order modulations of near-linear dispersive wave trains with an application to plasma waves
An asymptotic procedure is proposed to describe the slow modulations of dispersive wave trains when the dispersive effects and the nonlinear distorsion are of the same order of magnitude. For a model equation, the system of modulation equations is derived up to the second order. At this order of approximation, it is seen that the dispersion relation includes partial derivatives not only of the amplitude but also of the wave vector components. Under some assumptions, a partial differential equation is obtained for the complex amplitude. This equation reveals common features with the modified Korteweg-de Vries equation and with a nonlinear Schroedinger equation. At the first order, it reduces to the cubic Schroedinger equation which had been directly obtained by several authors. Finally, the theory is applied to a plasma wave example.