近几十年来，玻色-爱因斯坦凝聚吸引了极大关注，并意外地为研究非线性孤波提供了一个极好的平台，如一维暗孤子、二维涡旋与三维涡环。其中一个前沿是研究多组分系统中的复合孤波，如二组分暗亮孤子。除了新颖的结构外，复合孤子还具有单组分系统不具有的独特动力学。该项目的主要研究目的是在平均场理论的框架下使用最新的计算技术并结合可能的理论分析进一步探索各种维度中复合孤子的性质。具体来说，我们在一维（1D）中将系统地研究从线性极限下构造复杂复合孤子，这些结构中包含众多孤子，在二、三组分系统中都还未系统研究。在2D中，相关的结构也将以类似的方式进行研究。接着，我们最近发现当以弱外力驱动涡旋明亮的亮组分时，涡旋明亮会做有趣的匀速圆周运动。我们将详细研究这些动力学的特征。在3D中，很多基本孤子的性质还亟待研究。我们计划研究涡线明亮，涡环明亮以及更奇特的Skyrmion结构。该研究将显著提高对多组分复合孤波的理解。

Bose-Einstein condensates have attracted much attention in the past several decades, and also unexpectedly provided an excellent playground for investigating nonlinear solitary waves, including, e.g., dark solitons, vortices, and vortex rings in one-, two-, and three-dimensional settings, respectively. One frontier of research revolves around composite solitary waves in multi-component systems with the dark-bright soliton as a prototypical structure, existing remarkably in fully repulsive condensates. In addition to the novel structures, they also possess interesting and unique dynamics not present in a single-component system. For example, two dark solitons, one in each component, can undergo beating or breathing dynamics. The main purpose of this proposal is to explore further multi-component solitons in various dimensions using state-of-the-art computational techniques, combined with theoretical analysis when possible, in the framework of the mean-field Gross-Pitaevskii equation. Specifically, we study systematically more complex composite structures containing many solitons stemming from their linear limits in a parabolic trap in one dimension (1D) in two-component and even three-component systems. Pertinent structures in 2D will also be studied in a similar manner. Next, when driving the bright component of a vortex-bright (VB) structure with a weak external force in a homogeneous setting, the VB remarkably undergoes periodic circular oscillations. The properties of the dynamics, e.g., the dependence of the oscillation radius and period on the force strength, will be examined in detail. In 3D, even the basic properties of many elementary solitons have not been studied. In this project, we would like to study the properties of a number of 3D structures such as the vortex-line-bright (VLB), the vortex-ring-bright (VRB), and the more exotic skyrmion structures in both isotropic and anisotropic traps. These efforts will significantly improve the understanding of composite solitary wave dynamics of multi-component condensates.