Transverse instability of electron-acoustic solitons in a relativistic degenerate astrophysical magnetoplasma

We study the nonlinear theory of small-amplitude electron-acoustic solitons (EASs) in a relativistic astrophysical magnetoplasma consisting of two-temperature electrons: a sparse population of relativistic nondegenerate classical electrons and a group of fully degenerate dense relativistic electrons (main constituent) immersed in a static magnetic field with a neutralizing stationary ion background. By using the multiple-scale reductive perturbation technique with the Lorentz transformation, the Zakharov-Kuznetsov (ZK) and the modified Zakharov-Kuznetsov (mZK) equations are derived to describe the evolution of EASs in two different regimes of relativistic degeneracy: $r_{d0}<50$ and $r_{d0}\gtrsim50$. The characteristics of the plane soliton solutions of ZK and mZK equations and the soliton energy are studied. We show that the solitons moving at an angle $\alpha$ to the external magnetic field can be unstable under transverse long-wavelength perturbations. The growth rates of instabilities are obtained and analyzed with the effects of the relativity parameter $\beta_{\rm{cl}}=k_BT_{\rm{cl}}/m_ec^2$ and the degeneracy parameter $r_{d0}$, where $k_B$ is the Boltzmann constant and $T_{\rm{cl}}$ is the temperature of classical electrons. Interestingly, the ZK solitons, even if it is stable for the first-order perturbations, can be unstable in the second-order correction. Furthermore, while the first-order growth rates of perturbations for ZK solitons tend to vanish as $\alpha\rightarrow 38^\circ$, that for the mZK soliton goes to zero as $\alpha\rightarrow 90^\circ$. However, depending on the angle $\alpha$, the growth rates are found to be reduced either by increasing the values of $\beta_{\rm{cl}}$ or by decreasing the values of $r_{d0}$. The applications of our results to astrophysical plasmas, such as those in the environments of white dwarfs are discussed.