Magnetically Insulated Diode: Existence of Solutions and Complex Bifurcation. I

TL;DR

This paper analyzes the existence of solutions and complex bifurcations in a magnetic insulation vacuum diode model, providing new mathematical insights and numerical results for different insulation regimes.

Contribution

It introduces a novel reduction of the Vlasov-Maxwell system to nonlinear ODEs and explores bifurcations in the fully insulated diode regime.

Findings

Existence of solutions for the electric and magnetic fields in the insulation regime.

Identification of complex bifurcation structures depending on system parameters.

Determination of optimal diode spacing through bifurcation analysis.

Abstract

In order to avoid the electron oscillation of the cathode and enhance the work efficiency of a vacuum diode, an approach for analyzing the solutions and complex bifurcation has been proposed and used to determine the optimal trajectory of electron motion of the vacuum diode. This work is focusing on the stationary self-consistent problem of magnetic insulation in a space-charge-limited vacuum diode, modeled by a singularly perturbed 1.5-dimensional Vlasov-Maxwell system. We focus on the insulated regime, characterized by the reflection of electrons back toward the cathode at a point $x^{*}.$ The analysis proceeds in two primary stages. First, the original Vlasov-Maxwell system is reduced to a nonlinear singular system of ordinary differential equations governing the electric and magnetic field potentials. Subsequently, this system is further reduced to a novel nonlinear singular ODE for an effective potential $\theta(x).$ The existence of non-negative solutions to this final equation is established on the interval $[0, x^{*})$, where $\theta(x)>0$. This is achieved by reformulating the associated initial value problem into a system of coupled nonlinear Fredholm integral equations and proving the existence of fixed points for the corresponding operators. The most significant and previously unexplored case occurs when $\theta(x)<0$ on the interval $(x^{*}, 1]$, which corresponds to the fully insulated diode. For this regime, we present a novel numerical analysis of complex solution bifurcations, examining their dependence on system parameters and boundary conditions. Bifurcation diagrams illustrating the solution $\theta(x)$ as a function of the free boundary $x^{*}$ is constructed, and the insulated diode spacing is determined.