We present a semiconductor laser model which incorporates the gain bandwidth and nonlinear gain by using the multi-band microscopic theory of an electron - hole plasma in a semiconductor quantum-well medium. The approach is extremely robust, allowing us to take into account important many-body effects, as well as material and structural parameters of a given laser device. As a specific illustrative example, we resolve for the first time, the full multi-longitudinal mode and transverse filamentation instabilities of a master-oscillator power amplifier (MOPA) device.