  We study the spatio-temporal evolution of the nonlinear electrostatic
oscillations in a cold magnetized electron-positron (e-p) plasma using both
analytics and simulations. Using a perturbative method we demonstrate that the
nonlinear solutions change significantly when a pure electrostatic mode is
excited at the linear level instead of a mixed upper-hybrid and zero-frequency
mode that is considered in a recent study. The pure electrostatic oscillations
undergo phase mixing nonlinearly. However, the presence of the magnetic field
significantly delays the phase-mixing compared to that observed in the
corresponding unmagnetized plasma. Using 1D PIC simulations we then analyze the
damping of the primary modes of the pure oscillations in detail and infer the
dependence of the phase-mixing time on the magnetic field and the amplitude of
the oscillations. The results are remarkably different from those found for the
mixed upper-hybrid mode mentioned above. Exploiting the symmetry of the e-p
plasma we then explain a generalized symmetry of our non-linear solutions. The
symmetry allows us to construct a unique nonlinear solution up to the second
order which does not show any signature of phase mixing but results in a
nonlinear wave traveling at upper-hybrid frequency. Our investigations have
relevance for laboratory/astrophysical e-p plasmas.
