非線性薛丁格方程以及具位阻效應之新泊松−玻爾茲曼方程組漸近解的研究

Abstract

這篇碩士論文主要分成兩個主題,首先是非線性薛丁格方程 (Nonlinear Schrodinger Equation) 解的可壓縮極限問題之研究;第 二部分則是探討具位阻效應的泊松 - 玻爾茲曼方程組 (Poisson- Boltzmann Equation with steric effect) 的問題。 第一部份為初步探討兩道相向傳播的雷射光(counterpropagating optical beams)打在光折變晶體(photorefractive crystals)上的數 學模型,此類問題屬於一種非線性薛丁格方程。我們使用 [12] 的方 法,定義對應之能量泛函 H-function 來控制薛丁格方程解的質量密 度以及線性動量密度,進而描述其質量密度函數 (mass density) 與 線性動量 (linear momentum) 分別收斂到所對應的某類古典可壓縮 歐拉方程的解的質量密度以及線性動量,並解釋其現象。 第二部分是關於具位阻效應之新泊松-玻爾茲曼方程組的模型推 廣。這個模型是從 [10] 而來。因為加上離子半徑大小這個影響因 素,使我們能更精準的描述離子在離子通道中作用的情況,但在數 學的處理上也更加複雜。我們考慮兩種離子並且給定一些條件使這 個方程組具有完備性。此外,我們給定一些離子半徑和化學能的限 制,來和 [6] 以及 [7] 的模型做比較。最後我們得到上述的模型為我 們的一個特例。

In this thesis, we discuss two distinct types of partial differential equations. One is the nonlinear Schrodinger equation and the other is the Poisson-Boltzmann type equations. At first, we consider the system of two counterpropagating beams depicts the interaction of two counterpropagating optical beams in a photorefractive crystals. This is a preliminary study of nonlinear Schrodinger equation. We generalize the idea of [12] and define 'H ext{-function}' a modulated energy functional which may control the propagation of density and linear momentum in two-dimensional nonlinear Schrodinger equation. In the second scenario, we think over the behavior of ions in solutions with spatial effect, which depicts the behavior more accurate. In this work, we consider two-species ions PB_ns equation with steric effect which is subject to Robin type boundary condition. Which is more complicated than general PB equations. The main purpose of this part is to confirm the model is well posed and study the limiting behavior of the solution. Particularly, We can conclude that our model is more general than most of recent electrostatic models for electrolyte solutions.