Instability of Spatially — Periodic States for a Family of Semilinear PDE's on an Infinite Strip

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 spatially‐periodic 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 non‐autonomous 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.