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|∼γ5/2 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|∼γ2 is recovered.
J. Crawford, A. Jayaraman
Journal of Mathematical Physics