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 $\gamma$ vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude $\rho (t)$ is studied using an expansion in $\rho$. As $\text{glim}$, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: $\rho (t)\equiv {\gamma }^{2}r(\gamma t)$. This renders the theory finite and also indicates that the electric field exhibits trapping scaling $E\sim {\gamma }^2$. These singularities and scalings are independent of the specific $F_0(v)$ considered. The asymptotic dynamics of $r(\tau )$ depends on $F_0$ only through $\exp i\xi$ where $d{ϵ}_k\mathrm{/}dz=\mathrm{\mid }{ϵ}_k^{\mathrm{\prime }}\mathrm{\mid }\exp -i\xi \mathrm{/}2$. Similar results also hold for the electric field and distribution function.