氦四超流體具有孤子的解是眾所皆知的事情，在近期的研究裡，也可以觀測到在光學超晶格中超冷原子在系統內的動力學演化。在軟核玻色子的情況下也具有孤子的解，似乎是一件可以預期的事情，但是當玻色子之間的相互作用非常大時，硬核玻色子是否也具有孤子的解，就不是這麼顯而易見的事了，我們所要探討的主要問題就是尋找當硬核玻色子在光學超晶格裡與其他系統耦合時，是否能夠存在穩定輸送訊息的孤子解，這對工程與科學來說極為重要，因為真實系統皆為硬核玻色子與其他系統耦合，也有人論證真實物理世界下，硬核模型比軟核模型還精確。
本文第一章先簡介在各種物理系統裡面，孤子的基本數學模型及其由來。第二章中我們再來看看軟核玻色子在超流體中的孤子解，並得到在臨界速度時並不存在孤子解的結果。第三章我們要探討近兩三年來的研究成果，硬核玻色子在光學超晶格裡的孤子解，並且引入自旋相干態表象來研究此一問題，最後得出一個不直觀的結果，在臨界速度上還有亮孤子可穩定輸送訊息。第四章中，我們加入了另一類的費米子，並讓其費米子的自旋狀態與玻色氣體產生耦合，我們將探討以此類新穎的方法用在兩類氣體的耦合，能夠得到的結果為何，並點出此方法的局限性以及未來待解決的問題，我們還在裡面點出離散化的哈密頓量只要具有直接交換交互作用項，則可在凝聚態相變點附近存在孤子解，此為輸送問題最重要的關鍵點—存在可穩定輸送訊息的解。

It’s universally acknowledged that the Helium 4 superfluid has soliton solutions. In recent researches, we can also observe the dynamic evolutions of ultracold atoms in an optical superlattice system. It seems to be predictable that there are also soliton solutions in the condition of soft-core bosons, but if the interaction among bosons is quite huge, it will become tricky to ascertain if hard-core bosons also have solutions of soliton. Our topic is to find out if there are soliton solutions that stably transport messages when hard-core bosons couple with other systems in an optical superlattice. It’s critical to engineering and science, because all the real-life systems are the couplings of hard-core bosons and other systems, and there are also demonstrations which figure out that hard-core models are more accurate than soft-core models.
The first chapter of this paper will make a brief introduction to the basic models of soliton and their origins in a variety of mathematical models. In the second chapter we will see the soliton solutions of soft-core bosons in superfluid and prove that soliton solutions do not exist under critical speed. In chapter 3, a review toward the research results in the past 3 years will be performed, and we will use spin-coherent state representation to solve the problem-the soliton solutionsof hard-core bosons in optical superlattice. The result we get is not intuitive– there are bright solitons that can transport messages stably under critical speed. In chapter 4, we will add another kind of fermions, let their spin state couple with bose gases, investigate what result we will get when we use this novel way to couple two gases, and then point out the limitation of this way and problems waiting for solutions in the future. We will also show that as long as discrete Hamiltonian has direct exchange interaction term, it will be able to have soliton solutions near the phase transition point of condensed matter. This is the most critical point of transportation problem– solutions that can stably transport messages do exist.

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