Instability of Spatially — Periodic States for a Family of Semilinear PDE's on an Infinite Strip
✍ Scribed by Thomas J. Bridges; Alexander Mielke
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 895 KB
- Volume
- 179
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
A family of time-evolution equations, for which the time-independent part is a semilinear elliptic equation on an infinite strip, is considered with attention to the instability of spatiallyperiodic states. The analysis uses a sequence of center -manifold reductions; first, every local, small norm, bounded solution is contained in a center manifold. Second, it is proved that every local, small norm, solution of the linear stability problem is also contained in a related nonautonomous center manifold. For the case where the center manifold is foliated by periodic states, with wavelength L ( s ) , parametrized by s, the level sets of a suitably defined spatial Hamiltonian functional, we prove a geometric criterion for instability: a spatially periodic state is linearly unstable in the time-dependent problem if L'(s) > 0. Properties of the manifold of periodic states and their linear stability are explicitly constructed for a family of potentials.