Electron beam in a system with electrodes. I

The Pierce instability can arise and develop as an electron beam is transported between two electrodes between which a zero potential difference is maintained. The nonlinear dynamics of the electrons in a system of this type has been studied numerically. The Pierce instability can manifest itself in ''soft'' and ''hard'' regimes. In the soft case, the relaxation toward steady-state solutions can be described analytically. In the hard case, a virtual cathode, a spatial region from which electrons are reflected, forms in the system