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Boundary value problem of magnetically insulated diode: existence of solutions and complex bifurcation

Denis Sidorov, Alexander Sinitsyn, David Leguizamon, Liguo Wang

TL;DR

This work formulates and analyzes the boundary-value problem for magnetically insulated diodes by reducing a singularly perturbed Vlasov-Maxwell system to a nonlinear ODE framework for the electric and magnetic potentials. Introducing the effective potential $\theta(x)$, the authors establish existence of physically admissible solutions in the insulating regime and characterize the deflection point via a cubic relation that governs $\theta_d$ through a positive root $u$, with $\theta_d=u^2$. They prove existence and uniqueness of the reduced $D$-equation solution using Banach's fixed-point theorem and perform a detailed bifurcation analysis, including analytical and numerical construction of diagrams in terms of the parameters $\hat{k}$ and $\hat{\beta}$, revealing loop-like solution structures and multiple real roots. The study connects the mathematical structures to practical MID design, offering rules for stable operation, optimal spacing, and potential improvements in device performance and cost.

Abstract

The paper focuses on the stationary self-consistent problem of magnetic insulation for a vacuum diode with space-charge limitation, described by a singularly perturbed Vlasov-Maxwell system of dimension 1.5. The case of insulated diode when the electrons are deflected back towards the cathode at the point $x^{*}$ is considered. First, the initial VM system is reduced to the nonlinear singular limit system of ODEs for the potentials of electric and magnetic fields. The second step deals with the limit system's reduction to the new nonlinear singular ODE equation for effective potential $θ(x)$. The existence of non-negative solutions is proved for the last equation on the interval $[0, x^{*})$ where $θ(x)>0$. The most interesting and unexplored case is when $θ(x)<0$ on the interval $(x^{*}, 1]$ and corresponds to the case of an insulated diode. For the first time, a numerical analysis of complex bifurcation of solutions in insulated diode is considered for $θ(x)<0$ depending on parameters and boundary conditions. Bifurcation diagrams of the dependence of solution $θ(x)$ on a free point (free boundary) $x^{*}$ were constructed. Insulated diode spacing is found.

Boundary value problem of magnetically insulated diode: existence of solutions and complex bifurcation

TL;DR

This work formulates and analyzes the boundary-value problem for magnetically insulated diodes by reducing a singularly perturbed Vlasov-Maxwell system to a nonlinear ODE framework for the electric and magnetic potentials. Introducing the effective potential , the authors establish existence of physically admissible solutions in the insulating regime and characterize the deflection point via a cubic relation that governs through a positive root , with . They prove existence and uniqueness of the reduced -equation solution using Banach's fixed-point theorem and perform a detailed bifurcation analysis, including analytical and numerical construction of diagrams in terms of the parameters and , revealing loop-like solution structures and multiple real roots. The study connects the mathematical structures to practical MID design, offering rules for stable operation, optimal spacing, and potential improvements in device performance and cost.

Abstract

The paper focuses on the stationary self-consistent problem of magnetic insulation for a vacuum diode with space-charge limitation, described by a singularly perturbed Vlasov-Maxwell system of dimension 1.5. The case of insulated diode when the electrons are deflected back towards the cathode at the point is considered. First, the initial VM system is reduced to the nonlinear singular limit system of ODEs for the potentials of electric and magnetic fields. The second step deals with the limit system's reduction to the new nonlinear singular ODE equation for effective potential . The existence of non-negative solutions is proved for the last equation on the interval where . The most interesting and unexplored case is when on the interval and corresponds to the case of an insulated diode. For the first time, a numerical analysis of complex bifurcation of solutions in insulated diode is considered for depending on parameters and boundary conditions. Bifurcation diagrams of the dependence of solution on a free point (free boundary) were constructed. Insulated diode spacing is found.
Paper Structure (5 sections, 5 theorems, 8 equations, 4 figures)

This paper contains 5 sections, 5 theorems, 8 equations, 4 figures.

Table of Contents

  1. Motivation
  2. Setting of the problem
  3. Main Results
  4. Bifurcation diagrams
  5. Conclusion

Key Result

Proposition 3.1

Let $(u,v)$ be a solution of the IVP on $[0,\varepsilon)$ and define $\theta = u^{2} - 1 - v^{2}$. Then:

Figures (4)

  • Figure 1: Bifurcation diagram of the solutions of the cubic equation related to $u$ centered on the 3-multiplicity real solution
  • Figure 2: Bifurcation diagram of the solutions of the cubic equation related to $\theta$ centered on the 3-multiplicity real solution
  • Figure 3: Surface gotten from the parameters $\hat{k}$ and $\hat{\beta}$ for the solutions of the cubic equation
  • Figure 4: Surface gotten from the parameters $\hat{k}$ and $\hat{\beta}$ for the solutions of the stability of the $\theta$-based ODE

Theorems & Definitions (6)

  • Definition 3.1
  • Proposition 3.1
  • Theorem 3.1
  • Proposition 3.2: Child-Langmuir Law
  • Proposition 3.3: $\theta$ and $u$ solutions relationship
  • Proposition 3.4