  In this work, a systematic study, examining the propagation of periodic and
solitary wave along the magnetic field in a cold collision-free plasma, is
presented. Employing the quasi-neutral approximation and the conservation of
momentum flux and energy flux in the frame co-traveling with the wave, the
exact analytical solution of the stationary solitary pulse is found
analytically in terms of particle densities, parallel and transverse
velocities, as well as transverse magnetic fields. Subsequently, this solution
is generalized in the form of periodic waveforms represented by cnoidal-type
waves. These considerations are fully analytical in the case where the total
angular momentum flux $L$, due to the ion and electron motion together with the
contribution due to the Maxwell stresses, vanishes. A graphical representation
of all associated fields is also provided.
