The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification

摘要

We consider a spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification. We are interested in solutions bifurcating from the spatially homogeneous ground state in the case when pairs of imaginary eigenvalues simultaneously cross the imaginary axis. For this system we prove the bifurcation of a family of invariant tori which may contain quasiperiodic solutions. There is a serious difficulty in obtaining this result, because the linearization at the ground state possesses an essential spectrum up to the imaginary axis for all values of the bifurcation parameter. To construct the invariant tori, we use their invariance under the flow which manifests in a condition in PDE form. The nonlinear terms of this resulting PDE exhibit a loss of regularity. Since the linear part of this PDE is not smoothing, an adaption of the hard implicit function theorem (or Nash-Moser scheme) and energy estimates will be used to prove our result.