Traveling waves and pulses in a two-dimensional large-aspect-ratio system

Near a bifurcation point a system far from thermal equilibrium can be described by use of generalized Ginzburg-Landau equations. We present a systematic method to derive the nonlinear interaction terms of these equations in real space reflecting the selection rules as well as the stabilization of different patterns intrinsic in the basic equations of the systems under consideration. Our work treats the case of periodic instabilities of a homogeneous state in space as well as in time, where the interacting patterns are represented by traveling-wave trains having arbitrary orientations in a two-dimensional plane. Numerical solutions of two-dimensional pattern formation and wave propagation are presented using a system that allows for a backward Hopf bifurcation as is the case for the convection instability of a binary fluid mixture. The stability of the emerging traveling-wave structures is discussed in terms of phase-diffusion equations.