Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode

Details Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode

1994-07-06

arxiv.org/abs/patt-sol/9407001v1

Phys. Rev. Lett. 73, 656 (1994)

Nonlinear Sciences

Pattern Formation and Solitons

11 pages (Latex/RevTex), 2 figures available in hard copy from the Author (jdc@minerva.phyast.pitt.edu); paper accepted by Phy

Scientific paper

10.1103/PhysRevLett.73.656

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the two-dimensional unstable manifold of the equilibrium. The mode amplitude $\rho(t)$ decouples from the phase due to the spatial homogeneity of the equilibrium, and the resulting one-dimensional dynamics is analyzed using an expansion in $\rho$. As the linear growth rate $\gamma$ vanishes, the expansion coefficients diverge; a rescaling $\rho(t)\equiv\gamma^2\,r(\gamma t)$ of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling $|E_1|\sim\gamma^2$ as $\gamma\rightarrow0$. The dynamics for $r(\tau)$ depends only on the phase $e^{i\xi}$ where $d\epsilon_{{k}} /dz=|{\epsilon_{{k}}}|e^{-i\xi/2}$ is the derivative of the dielectric as $\gamma\rightarrow0$.

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