Modulation of electromagnetic waves in a relativistic degenerate plasma at finite temperature

Abstract: We study the modulational instability (MI) of a linearly polarized electromagnetic (EM) wave envelope in an intermediate regime of relativistic degenerate plasmas at a finite temperature $(T\neq0)$ where the thermal energy $(K_BT)$ and the rest-mass energy $(m_ec^2)$ of electrons do not differ significantly, i.e., $\beta_e\equiv K_{B}T/m_{e}c^2\lesssim~(\rm{or}~\gtrsim) 1$, but, the Fermi energy $(K_BT_F)$ and the chemical potential energy $(\mu_e)$ of electrons are still a bit higher than the thermal energy, i.e., $T_F>T$ and $\xi_{e}=\mu_e/K_{B}T\gtrsim1$. Starting from a set of relativistic fluid equations for degenerate electrons at finite temperature, coupled to the EM wave equation and using the multiple scale perturbation expansion scheme, a one-dimensional nonlinear Sch{\"o}dinger (NLS) equation is derived, which describes the evolution of slowly varying amplitudes of EM wave envelopes. Then we study the MI of the latter in two different regimes, namely $\beta_e<1$ and $\beta_e>1$. Like unmagnetized classical cold plasmas, the modulated EM envelope is always unstable in the region $\beta_e>4$. However, for $\beta_e\lesssim1$ and $1<\beta_e<4$, the wave can be stable or unstable depending on the values of the EM wave frequency, $\omega$ and the parameter $\xi_e$. We also obtain the instability growth rate for the modulated wave and find a significant reduction by increasing the values of either $\beta_e$ or $\xi_e$. Finally, we present the profiles of the traveling EM waves in the form of bright (envelope pulses) and dark (voids) solitons, as well as the profiles (other than traveling waves) of the Kuznetsov-Ma breather, the Akhmediev breather, and the Peregrine solitons as EM rogue (freak) waves, and discuss their characteristics in the regimes of $\beta_e\lesssim1$ and $\beta_e>1$.