Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Universal trapping scaling on the unstable manifold for a collisionless
  electrostatic mode   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
$\gamma\rightarrow0$.

6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
2
The categories of this paper are: math
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th