Eckmann J.-P., Wayne C.E., Wittwer P.
Geometric Stability Analysis for Periodic Solutions
of the Swift-Hohenberg Equation
(434K, postscript600dpi)
ABSTRACT. In this paper we describe invariant geometrical structures in the phase
space of the Swift-Hohenberg equation in a neighborhood of its periodic
stationary states. We show that in spite of the fact that these states are only
marginally stable (i.e., the linearized problem about these states has
continuous spectrum extending all the way up to zero), there exist finite
dimensional invariant manifolds in the phase space of this equation which
determine the long-time behavior of solutions near these stationary solutions.
In particular, using this point of view, we obtain a new demonstration of
Schneider's recent proof that these states are nonlinearly stable.