Nonlinear and nonlocal dynamics of spatially extended systems: Stationary states, bifurcations and stability

Spatially extended systems with nonlocal dynamics (e.g. ferromagnetic resonance or current instability) of the type

with uε ⁿ will be studied near the soft-mode instability (wave number k_c ≠ 0) of a stationary and uniform state. An amplitude equation is derived within the framework of a multiple-scale perturbation theory. A particular example of this class of nonlocal dynamics is also treated numerically. As the main result we find that in contrast to the well-known supercritical bifurcation into a stable periodic state, the uniform state can bifurcate supercritically into a stationary state of an amplitude-modulated fast oscillation in space.