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A simplified semiconductor laser model that is sufficient to describe laser dynamics on a picosecond time scale is presented. The model prescribes a time-domain differential equation regarding the complex amplitude of the light field. The simplification comes from an approximation of the semiconductor optical gain dynamics that typically exhibits two distinct time scales with a slow component on the order of hundreds of picoseconds and a fast component on the order of a few picoseconds or subpicoseconds. For laser dynamics concerning mainly the picosecond time scale, the slow component of the material gain dynamics can be treated as a static effect and the fast component can be treated as instantaneous. Such treatment also shines light on relations between semiconductor optical amplifier nonlinearities such as self-phase modulation and device parameters such as the linewidth enhancement factor and confinement factors. The model correctly predicts various picosecond laser dynamics of semiconductor lasers including the preference of a constant-power, FM mode-locking operation once the correct chirp condition is first established. A coupled multispatial mode (MSM) model based on the simplified time-domain laser model is used to successfully explain self-start AM mode-locking operations of single-contact Fabry-Perot laser diodes (FP-LDs) that have been observed experimentally. A mechanism for establishing the needed condition for single-contact FP-LD's self-FM mode-locking operations is also proposed based on the understanding of the MSM AM mode-locking operations.