In this chapter we remove the assumption made up to now—that the solutions are spatially periodic along the z-axis. We saw in Section II.3 that, for any fixed Reynolds number above criticality, there is an interval of axial wave numbers (mainly denoted by k in this chapter, instead of α) such that the perturbations of the linearized system grow exponentially (see Figure VII.1(b)). This is a major source of difficulty for our problem, since the method we used until now, crucially depended on the discreteness of the set of eigenvalues of the linearized operator, which precisely results from the h-periodicity in z (as shown in Figure II.5). For about 20 years, physicists have overcome this difficulty by considering slow modulations in space of the amplitudes of critical modes. The envelope equation they obtain is usually called the Ginzburg-Landau (G-L) equation (see Newell and Whitehead [New-Wh], Segel [Seg], and DiPrima et al. [DP-Ec-Seg]). We show in Section VII.3 how to obtain this complex partial differential equation in the present context.
KeywordsPeriodic Solution Normal Form Imaginary Axis Couette Flow Center Manifold
Unable to display preview. Download preview PDF.