  Starting from the Vlasov-Maxwell equations describing the dynamics of various
species in a quasi-neutral plasma, an exact relativistic hydrodynamic closure
for a special type of water-bag distributions satisfying the Vlasov equation
has been derived. It has been shown that the set of equations for the
macroscopic hydrodynamic variables coupled to the wave equations for the
self-consistent electromagnetic field is fully equivalent to the Vlasov-Maxwell
system.
  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, has been derived.
  Using the method of formal series of Dubois-Violette, a traveling wave
solution of the derived set of coupled nonlinear Schrodinger equations in the
case of circular wave polarization has been obtained. This solution is
represented as a ratio of two formal Volterra series. The terms of these series
can be calculated explicitly to every desired order.
