格若士-比塔烏斯基方程式的行進波解之穩定性研究

Abstract

格若士-比塔烏斯基方程式是現在用來描述弱耦合玻色氣體的主要方程式，其中用散射長度取代剛球半徑。散射長度有正有負，代表原子間相斥或相吸。極低溫下玻色氣體產生玻色-愛因斯坦凝聚態，其具有奇特性質，對基礎研究有重要意義，而且精密測量技術上也有很高的應用價值。現在凝聚態物理學也已經是物理學中的主流，吸引了許多優秀人才投入。本計畫將從對這些凝聚態性質之數學模型方程式上進行研究探討，針對其表現出的超流現象之進行波來作著研。我們將要發展一適當的數值計算方法求解此非線性格若士-比塔烏斯基方程式的進行波解，並檢驗其穩定性。再者，要對二維空間的渦流線與三維空間的渦流管現象進行模擬，期望能再進而發展一平行的計算演算法來作更高效率的求解與模擬。 表

In this project, we hope to develop a suitable numerical method to deal with a nonlinear Gross-Pitaevskii equation (GP). We would like to solve traveling waves of GP and examine the stability. The spectra of the linearized operators around these traveling waves are intimately connected to stability properties of the traveling waves, and to the long-time dynamics of solutions of GP. We hope to study these spectra in detail, both analytically and numerically. Moreover, we would like to numerically study the vortex line of traveling waves in two dimensions and the vortex tube of traveling waves in three dimensions. Furthermore, we expect to develop a parallel computing algorithm for high efficient solving traveling waves in high dimensions. 表