### Abstract

The weakly nonlinear description of 1D pattern forming systems in extended domains when the group velocity of the almost neutral wavetrains is not small requires to consider simultaneously two effects of different asymptotic order: the dominant group velocity advection and the necessarily smaller diffusion/dispersion. In this talk I will consider the amplitude equations that describe the dynamics of the system near onset in the reflection symmetric case, and the essential role that the boundary conditions play in this generic situation. I will review the simpler sub-models (hyperbolic and averaged equations) that can be derived depending on the relative size of the large domain length and the small supercriticallity, and I will also examine the interaction of these fastly advected wavetrains with a mean field, when a slow large scale mode is present at onset.