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The impact of the wavelength-dependent properties of quantum-well (QW) saturable-absorbers (SAs) on the mode-locked operation of monolithic two-section QW lasers is examined with a travelling-wave description that includes a model for the optical response of the active material at finite temperatures. Good-quality mode-locking regimes occur only in small regions of parameter space due to the wavelength-dependent absorption and saturation energy of the SA section. Joule heating associated with photocurrent generation in the SA section dynamically modifies the relative detuning, and may eventually lead to the disappearance of mode-locking. Either proper thermal management of the absorber section or absorbers with wavelength-independent properties are needed in order to improve the breadth of the regime of good-quality mode-locked operation.