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

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 not equal 0) where the thermal energy (KBT) and the rest-mass energy (m(e)c(2)) of electrons do not differ significantly, i.e., beta(e) = KBT/m(e)c(2) less than or similar to (or greater than or similar to) 1, but the Fermi energy (KBTF) 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)/KBT greater than or similar to 1. 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 Schodinger (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) less than or similar to 1 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) less than or similar to 1 and beta(e) > 1.