A two-dimensional electrodynamic model is used to study particle acceleration and non-thermal emission mechanisms in the pulsar magnetospheres. We solve distribution of the accelerating electric field with the emission process and the pair-creation process in meridional plane, which includes the rotational axis and the magnetic axis. By solving the evolutions of the Lorentz factor, and of the pitch angle, we calculate spectrum in optical through $gamma$-ray bands with the curvature radiation, synchrotron radiation, and inverse-Compton process not only for outgoing particles, but also for ingoing particles, which were ignored in previous studies. We apply the theory to the Vela pulsar. We find that the curvature radiation from the outgoing particles is the major emission process above 10 MeV bands. In soft $gamma$-ray to hard X-ray bands, the synchrotron radiation from the ingoing primary particles in the gap dominates in the spectrum. Below hard X-ray bands, the synchrotron emissions from both outgoing and ingoing particles contribute to the calculated spectrum. The calculated spectrum is consistent with the observed phase-averaged spectrum of the Vela pulsar. We show that the observed five-peak pulse profile in the X-ray bands of the Vela pulsar is reproduced by the inward and outward emissions, and the observed double-peak pulse profile in $gamma$-ray bands is explained by the outward emissions.
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