Amplitude equations for electrostatic waves: Universal singular behavior in the limit of weak instability

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the unstable manifold of the equilibrium F 0(v). The mode eigenvalue arises from a simple zero of the dielectric ε k (z); as the linear growth rate γ vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude ρ(t) is studied using an expansion in ρ. As γ→0+, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: ρ(t)≡γ2 r(γt). This renders the theory finite and also indicates that the electric field exhibits trapping scaling E∼γ2. These singularities and scalings are independent of the specific F 0(v) considered. The asymptotic dynamics of r(τ) can depend on F 0 only through exp iξ where dε k /dz=‖ε k ’ ‖exp−iξ/2. Similar results also hold for the electric field and distribution function.