Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow
Open Access
- 9 December 1985
- journal article
- Published by The Royal Society in Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- Vol. 402 (1823) , 299-322
- https://doi.org/10.1098/rspa.1985.0120
Abstract
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 J. T. Stuart and R. C. DiPrima (Proc R. Soc. Lond. A 362, 27-41 (1978)). The centre 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.Keywords
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