We develop a monolithic picosecond laser pulse generator, based on the classical design of a group-III-nitride Fabry-Pérot laser diode with electrically separated ridge sections. We use two different multi-section design variants, with the absorber section placed either in the center or at the end of the ridge. Profiting from the very low lateral conductivity in the p-type GaN top contact layer, we implement the multi-section concept just by etching off small sections of the top metalization on the ridge. The physical mechanism underlying short pulse generation within such system, operating in the 400 - 435nm wavelength range, strongly depends both on the reverse bias applied to the absorber and the forward current in the gain section. Varying the applied reverse bias affects both the absorption and the carrier lifetime in the absorber section through changes in the QW internal field. In consequence we can distinguish between different modes of operation. For moderately long carrier lifetimes the absorber stabilizes relaxation oscillations in the GHz frequency range and self-pulsation occurs, of relatively long duration. With increasing reverse bias, and thus decreasing carrier lifetime, we observe a transition to self-Q-switching. Finally, at large enough negative bias, the carrier life time in the absorber is so short that the laser diode operates in a passive self-mode-locking regime with a repetition rate of 87 GHz and pulse duration of 2 ps for a cavity length of 540 μm.