兩分量玻色─愛因斯坦凝聚中孤立子碰撞的數值模擬

Abstract

我們用數值方法來模擬在一維及二維空間兩分量玻色-愛因斯坦凝聚中二孤立子(soliton) 的碰撞，探討孤立子在交互作用時速度和形狀的變化。我們用一個梯度下降法[2](gradient flow method) 來計算二維空間中孤立子的形狀，以及利用時間分步正弦擬譜法[1](time-splitting sine pseudospectral method) 來計算波函數隨時間的變化。數值模擬的結果顯示在一維空間中若孤立子間若有足夠強的相斥作用力，則它們的碰撞像是彈性碰撞；而在強相吸作用力下，孤立子在碰撞後將分為兩個或多個波包(wave packets)。在二維空間中，孤立子間的相吸作用力如果夠強，則會在碰撞過程中發生爆破現象(blow up phenomenon)，其它的情形下孤立子將在碰撞後成為漸漸散開(spread out) 的波包。

We investigate interaction of bright solitons for two-component Bose-Einstein condensates (BECs) in one and two dimensions numerically (1D, 2D). The numerical methods we adopt are: (1) Gradient flow with discrete normalization (GFDN) method for computing the profile function of solitons in 2D. We use backward Euler sine pseudospectral (BESP) method to discretize it. The algorithm is constructed by Chern and Bao [2]. (2) Timesplitting sine pseudospectral (TSSP) method for computing the evolution of wave functions. The algorithm is construct by Bao [1]. We discuss the change of velocities and shapes of the wave packets during and after the interactions between them. It is found that (1) In 1D, soliton collisions are like elastic collisions under strong repulsive interactions. When the interactions are attractive and strong enough, the wave packets may split into two or more parts after collisions. (2) In 2D, wave packets spread out after collisions when the interactions are repulsive or weak attractive. The wave functions blow up during interactions when the attractive interactions are strong enough.