论文标题
磁滞和线性稳定性分析对带有空间相互作用的Poisson的多个稳态解决方案
Hysteresis and Linear Stability Analysis on Multiple Steady-State Solutions to the Poisson--Nernst--Planck equations with Steric Interactions
论文作者
论文摘要
在这项工作中，我们在数值上研究了多个稳态溶液对具有差异性边界条件（适用于离子通道的dirichlet边界条件）的一种空间泊松（nernst-planck（pnp）方程）的线性稳定性。有了数值发现的多个稳态溶液，我们获得了$ S $形的电流 - 电压和电流浓度曲线，显示离子电导对电压和边界浓度的滞后响应具有内存效应。提出了边界价值问题来定位分叉点并预测$ S $形曲线上分叉点附近的局部分叉图。开发了线性稳定性分析的数值方法，以了解仅在数值上可用的稳态解决方案的稳定性。提出了有限差异方案来解决涉及差异操作员的派生特征值问题。线性稳定性分析表明，$ S $形的曲线具有两个线性稳定的分支不同的电导水平和一个线性不稳定的中间分支，表现出经典的双重性磁滞。正如线性稳定性分析中预测的那样，从不稳定分支上的稳态解决方案到稳定分支上的一个稳态解决方案的过渡动力学取决于与主要特征值模式相关的扰动。进一步的数值测试表明，线性稳定性分析中提出的有限差异方案是二阶精确的。在这项工作中开发的数值方法可以应用于研究其稳态解决方案周围计算的一类时依赖性问题的线性稳定性。
In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson--Nernst--Planck (PNP) equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain $S$-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the $S$-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the $S$-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis, transition dynamics, from a steady-state solution on the unstable branch to a one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.
