Microwave Soliton Generation and Propagation in a Cylindrical Partially Plasma Filled Waveguide

A perfect electrical conductor (PEC) cylindrical waveguide with a radius of r$_{1}$ is considered as the main part of a basic microwave generator model in the cylindrical geometry which supports the axial movement of an electron beam of radius r$_{2 }\le $ r$_{1}$ inside it. Using fluid theory of plasma, it has been shown that this structure is capable of supporting nonlinear Schr\"{o}dinger (NLS) soliton generation and propagation. The wave equation for the vector potential \textbf{A} has been derived using plasma dispersion relation. The equation has then been separated into two different equations in the transverse and axial directions, considering a solution in the form of a function of transverse variables multiplied by another time dependent function of axial variable, the latter itself having two components: a fast oscillation with a slowly varying amplitude. Once $\beta $, the propagation constant in the axial direction, obtained by applying boundary conditions in the former equation, the latter equation can be manipulated by imposing a perturbation on the dielectric constant. This will result in a perturbation on propagation constant which in turn will induce to the calculations the nonlinear term required for NLS equation in the form of a pondermotive force, completing derivation of the required NLS equation supporting the soliton formation and propagation.