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., $β_e\equiv K_{B}T/m_{e}c^2\lesssim~(\rm{or}~\gtrsim) 1$, but, the Fermi energy $(K_BT_F)$ and the chemical potential energy $(μ_e)$ of electrons are still a bit higher than the thermal energy, i.e., $T_F>T$ and $ξ_{e}=μ_e/K_{B}T\gtrsim1$. 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{ö}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 $β_e<1$ and $β_e>1$. Like unmagnetized classical cold plasmas, the modulated EM envelope is always unstable in the region $β_e>4$. However, for $β_e\lesssim1$ and $1<β_e<4$, the wave can be stable or unstable depending on the values of the EM wave frequency, $ω$ and the parameter $ξ_e$. We also obtain the instability growth rate for the modulated wave and find a significant reduction by increasing the values of either $β_e$ or $ξ_e$. 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 $β_e\lesssim1$ and $β_e>1$.
Modulational instability and nonlinear evolution of two-dimensional electrostatic wave packets in ultra-relativistic degenerate dense plasmas
研究了超相对论（UR）简并稠密电子-离子等离子体中静电波包的非线性传播，其动力学过程由非局域二维非线性类薛定谔方程控制.然后利用耦合方程组研究了均匀波列对多维形式的微扰的调制不稳定性。得到了MI的条件，并证明了与简并电子UR压力相关的无量纲参数$β\proptoλ_C n_0^{1/3}$（$λ_C$是约化康普顿波长，$n_0$是粒子数密度）使$n_{0}\sim10^{30}$ cm $^{-3}$处的稳定（不稳定）区域向更高密度处的不稳定（稳定）区域（即$n_{0}\gtrsim7\times10^{33}$）移动.研究还发现，$n_{0}$的值越高，调制波数越低时截止MI的增长率越低。此外，数值研究了波包的动力学演化。我们表明，它们要么分散或爆破在有限的时间内，当波的作用是低于或高于阈值。这些结果对于理解UR简并致密等离子体中调制波包的性质及其多维演化，例如白矮星{和/或}前超新星恒星内部的调制波包的性质具有重要意义.
