Selection of scales in pattern-forming dynamics

In the strongly nonlinear regime many pattern-forming systems, such as premixed flames, film flows, and three-dimensional hydrodynamical flows, exhibit a remarkable nonlinear phenomenon: the resulting patterns are substantially longer than the wavelength of the linearly most unstable mode. Usually such an inverse cascade, or coarsening is attributed to high-order nonlinear effects. We show, however, that the coarsening may be well described by a proposed weakly nonlinear evolution equation. The key of the model is the dispersion relation, which, being the kernel of Fourier convolution operator, captures the essential properties of strong instabilities in nonlinear systems.