Classical solutions for a van Roosbroeck–Helmholtz model for a semiconductor laser diode

This work presents the first rigorous mathematical analysis of a coupled light–matter model for semiconductor laser diodes, combining the transient van Roosbroeck system for charge transport with a Helmholtz eigenvalue problem describing the optical field. The paper proves local existence and uniqueness of solutions under physically relevant assumptions by establishing the local Lipschitz continuity of the nonlinear stimulated recombination operator and applying the abstract theory of quasi-linear parabolic equations. The results provide a mathematically consistent framework for semiconductor laser models that incorporate the interaction between charge transport and optical modes through stimulated emission.