Amplitude Equations for Electrostatic Waves: multiple species

Details Amplitude Equations for Electrostatic Waves: multiple species Amplitude Equations for Electrostatic Waves: multiple species

1997-06-10

arxiv.org/abs/patt-sol/9706001v1

J. Math. Phys., 39 (1998) 4546

Physics

Condensed Matter

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

Scientific paper

10.1063/1.532635

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.

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