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 amp litude equations describing the interaction between counterpropagating wave s. Dissipation, detuning and forcing are all assumed to be weak and any cou pling to mean fields (such as large scale flows in fluid systems) is ignore d. If the aspect ratio is moderately large the system is described by a pai r of nonlocal equations for the (complex) amplitudes of the waves. The dyna mics 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 investigate d as a function of the driving in three particular cases. (C) 2000 Elsevier Science B.V. All rights reserved.