Amplitude Equations for Electrostatic Waves: multiple species

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. γ → 0+ where γ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of A(t) on γ. Generically the scaling |A(t)| = γ5/2r(γt) as γ → 0+ is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling |Ek| ∼ γ 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 |Ek| ∼ γ 2 is recovered. Typeset using REVTEX 1