Modulational instability and nonlinear evolution of two-dimensional electrostatic wave packets in ultra-relativistic degenerate dense plasmas

Abstract

We consider the nonlinear propagation of electrostatic wave packets in an ultra-relativistic (UR) degenerate dense electron-ion plasma, whose dynamics is governed by the nonlocal two-dimensional nonlinear Schr{ö}dinger-like equations. The coupled set of equations are then used to study the modulational instability (MI) of a uniform wave train to an infinitesimal perturbation of multi-dimensional form. The condition for the MI is obtained, and it is shown that the nondimensional parameter, $\beta \propto {\lambda }_C{n}_0^{1/3}$ (where ${\lambda }_C$ is the reduced Compton wavelength and $n_0$ is the particle number density), associated with the UR pressure of degenerate electrons, shifts the stable (unstable) regions at $n_0\sim 10^{30}$ cm${}^{-3}$ to unstable (stable) ones at higher densities, i.e. $n_0\gtrsim 7\times 10^{33}$. It is also found that {the} higher the values of $n_0$, the lower is the growth rate of MI with cut-offs at lower wave numbers of modulation. Furthermore, the dynamical evolution of the wave packets is studied numerically. We show that either they disperse away or they blowup in a finite time, when the wave action is below or above the threshold. The results could be useful for understanding the properties of modulated wave packets and their multi-dimensional evolution in UR degenerate dense plasmas, such as those in the interior of white dwarfs {and/or} pre-Supernova stars.

11 pages, 7 figures; Revised version to appear in Physics of Plasmas (April issue, 2011)

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