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We experimentally study the pulse generation and the pulse train timing and amplitude stability of a monolithic passively mode-locked multisection quantum-well semiconductor laser. The laser emits a stable optical pulse train at a wavelength of 1070 nm and at a fundamental repetition rate of 20 GHz. Timing jitter and amplitude jitter analysis allows identifying distinct laser operating regimes of stable picosecond optical pulses. We obtain an ultra-low pulse-to-pulse timing jitter of 22 fs and find emission regimes which are unaffected by amplitude jitter. The lowest pulse width amounts to 2.6 ps, and the highest pulse peak power is estimated to be 556 mW at a low timing jitter of 170 fs and a negligible relative amplitude jitter of 0.05. [ABSTRACT FROM AUTHOR]