Stability of traveling waves for a conserved field

The stability of traveling waves is investigated for a model describing the post threshold behavior beyond oscillatory instabilities for a class of systems with a conserved order parameter. Oscillatory instabilities and their post threshold behavior in systems with unconserved order parameters have been a central topic of nonlinear science during the recent decade. The most famous equation in this context is the Ginzburg-Landau equation with complex coefficients. Here I discuss a straight forward generalization of this Ginzburg-Landau equation which covers also spinodal decomposition and the crossover to an oscillatory instability for a globally conserved order parameter. Especially, the modification of the border to spatiotemporal chaos is considered, which is described by the so-called Benjamin-Feir resonance.