Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability

Abstract

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 $\glim$, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: $ρ(t)\equivγ^2\,r(γt)$. This renders the theory finite and also indicates that the electric field exhibits trapping scaling $E\simγ^2$. These singularities and scalings are independent of the specific $F_0(v)$ considered. The asymptotic dynamics of $r(τ)$ depends 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.