The dynamics of an electron–ion plasma is studied when a mono-energetic relativistic electron beam penetrates the expanding plasma. Combined effects of thermal pressure and ambipolar electrostatic potential are considered for fully relativistic multi-fluids model where the quasi-neutrality assumption is used for a beam–plasma system. Relativistic effects are considered for both density and velocity of the beam fluid. Ion acceleration is depicted through a spike-like structure resulting from non-local charge separation and associated with the beak-down of quasi-neutrality. The beam initial speed is found to enhance the spike amplitude and change its position. Moreover, relativistic effects are particularly found significant for a non-thermal plasma.

We study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

The self-focusing of an intense right circularly polarized Gaussian laser pulse in magnetized plasma is studied. The ions are taken to be immobile and relativistic mass effect is incorporated in both the plasma frequency (ωp) and the electron cyclotron frequency (ωc) while determining the ponderomotive force on electrons. The ponderomotive force causes electron expulsion when the effective electron cyclotron frequency is below twice the laser frequency. The nonlinear plasma dielectric function due to ponderomotive and relativistic effects is derived, which is then employed in beam-width parameter equation to study the self-focusing of the laser beam. From this, we estimate the importance of relativistic self-focusing in comparison with ponderomotive self-focusing at moderate laser intensities. The beam width parameter decreases with magnetic field indicating better self-focusing. When the laser intensity is very high, the relativistic gamma factor can be modeled as ${\rm \gamma} = 0.8\left({{{{\rm \omega} _c } / {\rm \omega} }} \right)+ \sqrt {1 + a_0^2 }$$\text{γ}=0.8({\text{ω}}_c/\text{ω})+\sqrt{1+a_0^2}$ where ω and a0 are the laser frequency and the normalized laser field strength, respectively. The cyclotron effects on the self-focusing of laser pulse are reduced at high field strengths.

We consider ion acceleration at the front surface of overdense plasma by a short laser pulse. In this configuration, the laser ponderomotive pressure pushes the background electrons, and a double layer is produced at the boundary of the overdense region. The ions are accelerated by the electrostatic field of the double layer. If the laser intensity is so large that the plasma becomes relativistically transparent, then ion trapping in the running double layer and acceleration to relativistic energies is possible. We study this physics using one-dimensional particle-in-cell simulations. Ion acceleration in one- and two-component plasmas is considered. It is shown that the proton acceleration is more effective when they represent only a small dope to the heavy background ions.