Magnetically insulated diode: existence of solutions and complex bifurcation. I

This paper studies magnetic insulation in a space-charge-limited vacuum diode through a stationary self-consistent model derived from a singularly perturbed 1.5-dimensional Vlasov-Maxwell system. The central objective is to characterize the transition to the insulated regime, in which electrons are reflected toward the cathode at a free boundary point [Formula: see text]. The analysis is developed in two stages. First, the original kinetic model is reduced to a nonlinear singular system for the electric and magnetic potentials, and then to a nonlinear singular equation for the effective potential [Formula: see text]. For the region [Formula: see text], where [Formula: see text], we prove the existence of physically admissible nonnegative solutions by reformulating the problem as a coupled system of nonlinear Fredholm integral equations and establishing fixed-point existence. Second, for the fully insulated regime [Formula: see text], where [Formula: see text], we perform a bifurcation analysis of complex solutions and their dependence on system parameters and boundary conditions. The resulting bifurcation diagrams identify critical parameter thresholds, describe regime transitions, and provide a quantitative estimate of the insulated diode spacing. These results provide an integrated analytical-computational approach for predicting magnetic-insulation behavior in high-power vacuum diodes for the reduced model studied, combining rigorous existence results with computational bifurcation validation and parameter-space exploration.