Reduction of the Longitudinal Multi-Modes Dynamics of a Driven Semiconductor Amplifier to a Single Scalar Partial Differential Equation

In the plane wave approximation, we analytically study spatio-temporal dynamics of a semiconductor amplifier driven by a coherent injected field in a Fabry-Perot configuration. We manage to reduce the dynamics to a single evolution equation for the slow variable represented by the carrier density, to analytically compute the stationary field configurations and to predict their stability. The numerical simulations, performed by implementing an efficient and accurate split-step code, perfectly agree with the analytical results.