Amplitude Equations for Electrostatic Waves: multiple species

Abstract

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.

41 pages (Latex/RevTex), 1 postscript figure included (psfig). Figure also available in hard copy from the authors

References in corpus (4)

• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
• Scaling and singularities in the entrainment of globally-coupled oscillators
• Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability
• Nonlinear saturation of electrostatic waves: mobile ions modify trapping scaling

Cited by in corpus (1)

• First Principles Justification of a ``Single Wave Model'' for Electrostatic Instabilities