Dynamics of counterpropagating waves in parametrically forced, large aspect ratio, nearly conservative systems with nonzero detuning

Abstract

The dynamics of parametrically driven, slowly varying counterpropagating wave trains in nearly conservative systems are considered. The system is assumed to be invariant under reflection and translations in one direction, and periodic boundary conditions with period L are imposed, with L large but not too large in order that the effect of detuning be significant. The dynamics near the minima of the resulting resonance tongues are described by a system of coupled nonlocal Schrödinger equations with damping and parametric forcing. Elsewhere the long time behavior of the system is described by a damped complex Duffing equation with real coefficients, whose solutions relax to spatially uniform standing waves. Near the bicritical points where two adjacent resonance tongues intersect a pair of coupled damped complex Duffing equations captures the properties of both pure and mixed modes, and of the periodic solutions resulting from a Hopf bifurcation on the branch of mixed modes. As an application, we consider a Faraday system in an annulus in which a pair of counterpropagating surface gravity-capillary waves are excited parametrically by vertical vibration of the container, including the mean flow driven by time-averaged Reynolds stresses due to oscillatory viscous boundary layers along the bottom and the free surface. This mean flow is shown to have a large effect near the bicritical point, where the mean flow changes the dynamics of the system both quantitatively and qualitatively. In particular, inclusion of the mean flow permits Hopf bifurcations from the branches of pure standing waves, and parity-breaking bifurcations from mixed modes.