A new nonlinear electromagnetic wave mode in a plasma is reported. Its existence depends on the interaction of an intense circularly polarized electromagnetic wave with a plasma, where quantum electrodynamical photon–photon scattering is taken into account. As an illustration, we consider a pair plasma and show that the new mode can be significant in astrophysical settings and in the next generation of laser–plasma systems.

A theoretical explanation is given for tripolar electric field signals observed recently in the Earth's magnetosphere by the Cluster multi-spacecraft mission and in interplanetary space by the WIND mission, which feature an overall potential drop characteristic for weak double layers, with adjacent potential minima and maxima that trap both ions and electrons, respectively. A kinetic theory, based on Schamel's model for trapped particle distribution, is presented for electron and ion holes coupled with weak double layers, whose properties are studied analytically and numerically. In contrast to earlier models of double layers, which require external excitation in the form of asymptotically non-Maxwellian velocity distributions, tripoles only trap particles locally and the corresponding asymptotic particle distributions are Maxwellian. In the small amplitude limit, $e\phi\,{\ll}\, T_{\rm e}, T_{\rm i}$, such structures propagate with a speed that is in the ion thermal velocity range and they can only be excited if there is a relative drift between electrons and ions, whose speed is in the electron thermal velocity range. Conversely, non-propagating tripoles are found in the large amplitude regime and in the absence of particle streaming.

A review of a technique for the application of a new Lambert $W$ function for solving transcendental equations arising in classical problems of plasma and electron beam theories is presented, and the basic mathematical properties of the Lambert $W$ function are outlined. The problems discussed include equilibrium electric charge of a solitary body in plasma, nonlinear ion-acoustic waves, the equilibrium configuration of a neutralized electric beam, the dynamics of shaping virtual cathodes in a medium with viscous friction, and the study of the dispersion equation for electron plasma waves in the framework of generalized Boltzmann kinetics.

The nonlinear interaction, due to quantum electrodynamical effects, between photons is investigated using a wave-kinetic description. Starting from a coherent wave description, we use the Wigner transform technique to obtain a set of wave-kinetic equations, the so called Wigner–Moyal equations. These equations are coupled to a background radiation fluid, whose dynamics are determined by an acoustic wave equation. In the slowly varying acoustic limit, we analyse the resulting system of kinetic equations, and show that they describe instabilities, as well as Landau-like damping. The instabilities may lead to the break-up and focusing of ultra-high-intensity multi-beam systems, which in conjunction with the damping may result in stationary strong field structures. The results could be of relevance for the next generation of laser–plasma systems.

The effects of relativistic mass increase is considered in the context of intense laser–plasma interactions. It is found that the result of the relativistic effect is to enhance the self-compression and collapse of the intense laser pulse, making it possible to reach the Schwinger field limit, at which pair creation would need to be considered.

The effect of density and size of dust grains on the electron energy distribution function (EEDF) in low-temperature complex plasmas is studied. It is found that the EEDF depends strongly on the dust density and size. The behavior of the electron temperature can differ significantly from that of a pristine plasma. For low-pressure argon glow discharge, the Druyvesteyn-like EEDF often found in pristine plasmas can become nearly Maxwellian if the dust density and/or sizes are large. One can thus control the plasma parameters by the dust grains.

Two applications of the Noether method for fluids and plasmas are presented based on the Euler–Lagrange and Euler–Poincaré variational principles, which depend on whether the dynamical fields are to be varied independently or not, respectively. The relativistic cold laser–plasma equations, describing the interaction between an intense laser field with a cold relativistic electron plasma, provide a useful set of equations amenable to both variational formulations. The derivation of conservation laws by the Noether method proceeds from the Noether equation, whose form depends on the variational formulation used. As expected, the expressions for the energy–momentum conservation laws are identical in both variational formulations. The connection between the two Lagrangian densities is shown to involve the mass conservation and Lin constraints associated with the cold relativistic electron fluid.