Dynamics of counterpropagating waves in parametrically forced systems

Parametrically driven waves in weakly dissipative systems with one extended dimension are considered. Multiple scale techniques are used to derive amplitude equations describing the interaction between counterpropagating waves. Dissipation, detuning and forcing are all assumed to be weak and any coupling to mean fields (such as large scale flows in fluid systems) is ignored. If the aspect ratio is moderately large the system is described by a pair of nonlocal equations for the (complex) amplitudes of the waves. The dynamics of these equations are studied both in annular and bounded geometries with lateral walls. The equations admit spatially uniform solutions in the form of standing waves and spatially nonuniform solutions with both simple and complex time-dependence. Transitions among these states are investigated as a function of the driving in three particular cases.