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.
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