Stability of solitary waves in a semiconductor drift-diffusion model

We consider a macroscopic (drift-diffusion) model describing a simple microwave generator, consisting of a special type of semiconductor material that, when biased above a certain threshold voltage, generates charge waves. These waves correspond to travelling wave solutions of the model equation which, however, turn out to be unstable in a standard formulation of the travelling wave problem. Here a different formulation of this problem is considered, where an external voltage condition is applied in the form of an integral constraint. Global existence of this novel Cauchy problem is proven and the results of numerical experiments are presented, which suggest the stability of solitary waves. In addition, a small amplitude limit is considered, for which linearized orbital stability of solitary waves can be proven.