基于Painlevé II的电扩散模型中的Bäcklund通量量子化
Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II
摘要：
重新考虑了一个先前建立的稳态一维双离子跨液接界电扩散模型。它涉及三个耦合的一阶非线性常微分方程，并在其核心的二阶Painlevé II方程。解现在通过贝克隆变换分组为无限序列，部分由两个Bäcklund不变量标记。每个序列的特征在于两个离子种类的均匀间隔的量子化通量，因此电流密度的均匀间隔的量子化。确定了有限的精确解，正离子浓度和量子化通量，从零电场普朗克发现的解决方案，并建议解释为基态加激发态的系统。当普朗克的电荷中性边界条件适用时，离子浓度的正性就成立了。精确的解决方案，获得了电场和离子浓度在搅拌良好的水库以外的每个面对的交界处，使制定更现实的边界条件。在近似形式中，这些导致Painlevé II的辐射边界条件。给出了说明性的数值解，并讨论了建立边界条件与通量量化序列结构的兼容性问题。
英文摘要：
A previously-established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential equations, and has the second-order Painlevé II equation at its core. Solutions are now grouped by Bäcklund transformations into infinite sequences, partially labelled by two Bäcklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current-density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck, and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck's charge-neutral boundary-conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary-conditions. In an approximate form, these lead to radiation boundary conditions for Painlevé II. Illustrative numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.
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