Generation and Propagation of Nonlinear Waves in Travelling Wave Tubes
  The generation and evolution of nonlinear waves in microwave amplifiers such as travelling wave tubes, free electron lasers and klystrons have been studied. The analysis is based on the hydrodynamic and field equations for the self-consistent evolution of the beam density distribution, the current velocity and the electromagnetic fields. A system of coupled nonlinear Schr\"{o}dinger equations for the slowly varying amplitudes of interacting beam-density waves has been derived. Under the approximation of an isolated mode neglecting the effect of the rest of the modes, this system reduces to a single nonlinear Schr\"{o}dinger equation for that particular mode. 
Stephan I. Tzenov