Amplitude Equations on Unstable Manifolds: singular behavior from neutral modes

Abstract

Perturbation theories that expand in the amplitudes of the unstable modes are an important tool for analyzing the nonlinear behavior of a weak instability which saturates in a final state characterized by small mode amplitudes. If the unstable mode couples to neutrally stable modes, such expansions may be singular because nonlinear effects are very strong even in the regime of weak instability and small amplitudes. Two models are discussed that illustrate this behavior; in each case the unstable mode corresponds to a complex conjugate eigenvalue pair in the spectrum of the linearized dynamics. In the first model, there is only a single neutral mode corresponding to a zero eigenvalue. This example is first solved exactly and then using amplitude expansions. The Vlasov equation for a collisionless plasma is the second model; in this case there are an infinite number of neutral modes corresponding to the van Kampen continuous spectrum. In each of the two examples, the neutral modes sharply reduce the size of the resulting nonlinear oscillation. For the Vlasov instability, the amplitude of the saturated mode is predicted to scale like γ², where γ is the linear growth rate.