在最優分割的分佈下之非線性特徵值問題

Abstract

在正的且大的排斥散射強度下，多種原子的玻色愛因斯坦凝聚現象之分佈，是本研究專題所關注的。在排斥散射強度趨向無窮大後，數學理論上已經有嚴格的證明，不同種類的原子玻色愛因斯坦凝聚現象分佈是呈現完全個別分開；在數值計算方面，模擬計算方法也已被發展了；同時觀察到一種現象:輪生結構。但是，當原子種類個數相當大後，計算結果是差的，甚至是不可靠。因此，本研究專題首要目標是發展有效地數值計算演算法，求解描述多種原子玻色愛因斯坦凝聚現象的非線性薛丁格方程系統。然後，找出原子種類個數越來越大時的分佈結構之準則。再者，我們希望探討到三維空間下的多種原子之分佈結構，並與二維空間下的情形進行比較。

In this project, we would like to study the distribution of segregated nodal domains of the mixture of Bose-Einstein condensates (BECs) under positive and large repulsive scattering lengths. It is shown that components of positive bound states may repel each other and form segregated nodal domains as the repulsive scattering lengths go to infinity. Numerical schemes, Gauss-Seidel-type iteration method and Jacobi-type iteration method, have been created to confirm the theoretical results. And a phenomenon, verticillate multiplying, is observed. But the numerical result is bad even unreliable when the number of segregated nodal domains of the mixture of BECs is large. Therefore, it is our first goal to develop an efficient numerical for simulating a multi-component BEC. Then, it is the second goal to find some principles in the distribution of segregated nodal domains of the mixture of multi-component BECs. Moreover, we hope to simulate multi-component BECs under a three dimensional domain and compare 2-dimensional and 3-dimensional cases.