Starting from the Vlasov-Maxwell equations describing the dynamics of various species in a free quasi-neutral plasma, an exact relativistic hydrodynamic closure for a special type of water-bag distributions satisfying the Vlasov equation has been derived. In the case of magnetized quasi-neutral plasma, the hydrodynamic substitution has been used to derive the hydrodynamic equations governing the evolution of the plasma density and the current velocity, which are coupled to the wave equations for the self-consistent electromagnetic fields.
Based on the method of multiple scales, a system comprising a vector nonlinear Schrodinger equation for the transverse envelopes of the self-consistent plasma wakefield, coupled to a scalar nonlinear Schrodinger equation for the electron current velocity envelope for free plasma, has been derived. In the case of magnetized plasma, it has been shown that the whistler wave envelopes of the three basic modes satisfy a system of three coupled nonlinear Schrodinger equations. Numerical examples for typical plasma parameters have been presented, which demonstrate the relevance of the results thus obtained to the so-called shock laser-plasma acceleration. In addition, it has been shown that in the case of magnetized plasma, the whistler waves facilitate the transverse confinement considerably.
For the case of classical vacuum polarization of Heisenberg-Euler type, a derivative nonlinear Schrodinger equation for the electromagnetic vector potential has been derived and its properties have been analysed. In the case, where an external constant magnetic field is applied, a similar derivative nonlinear Schrodinger equation has been derived. It has been shown that the external magnetic field considerably enhances the solitary wave formation. This important effect has been further discussed in view of a possible experiment called upon to observe the effect of vacuum polarization and magnetization at high intensity electromagnetic fields.