  The amplitude equation for an unstable electrostatic wave is analyzed using
an expansion in the mode amplitude $A(t)$. In the limit of weak instability,
i.e. $\gamma\to 0^+$ where $\gamma$ is the linear growth rate, the nonlinear
coefficients are singular and their singularities predict the dependence of
$A(t)$ on $\gamma$. Generically the scaling $|A(t)|=\gamma^{5/2}r(\gamma t)$ as
$\gamma\to 0^+$ is required to cancel the coefficient singularities to all
orders. This result predicts the electric field scaling $|E_k|\sim\gamma^{5/2}$
will hold universally for these instabilities (including beam-plasma and
two-stream configurations) throughout the dynamical evolution and in the
time-asymptotic state. In exceptional cases, such as infinitely massive ions,
the coefficients are less singular and the more familiar trapping scaling
$|E_k|\sim\gamma^2$ is recovered.
