Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

Details Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

1996-12-20

arxiv.org/abs/patt-sol/9612002v1

Mathematics

Functional Analysis

44 pages, plain tex, 0 figures

Scientific paper

10.1007/s002200050238

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.

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