Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

The stability conditions of plane waves against three-dimensional perturbations in plane Poiseuille flow, as described by a dispersive cubically nonlinear complex-amplitude equation, under perturbations quasi-periodic in two of the space dimensions are investigated. It is found that if the parameters satisfy certain conditions, a wave is totally stable. These conditions are an extension of those given for the lower dimensional case by Stuart and DiPrima (1978). The center manifold theorem is then used to investigate the nature of the solutions bifurcating from a marginally unstable plane wave. Hopf bifurcations occur in the 1, 2 or 3 perturbing sidebands that are neutrally stable to the unperturbed wave and can give rise to limit cycles or tori.