Multi{pulse Solutions to the Navier{stokes Problem between Parallel Plates

We consider the three-dimensional Poiseuille problem of a viscous incompressible uid ow between parallel plates. The ows under investigation are assumed to be traveling waves in streamwise direction with spatial periodicity 2==. In spanwise direction they are assumed to be uniformly close to the basic ow which enables us to use the spatial center{manifold reduction, where the spanwise variable takes the role of the time. For Reynolds numbers close to criticality the problem is reduced to a four-dimensional ODE whose lowest order terms coincide with the steady complex Ginzburg{Landau equation. Using perturbation arguments we relate reversible n{ pulse solutions of this equation to n{pulse solutions of the problem on a center manifold. Thus, we obtain multi{pulse solutions of the Navier{Stokes problem for parameters slightly below criticality. These solutions are localised in spanwise direction but periodic in streamwise direction.