在這篇論文，我們回顧了兩篇關於週期非線性薛丁格方程及系統的爆破解的論文。第一篇論文是由 T. Ogawa 教授和 Y. Tsutsumi 教授共同所寫的”Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition” [17] 而第二篇論文是由 F. Ivanauskas 教授和 G. Puriuškis 教授共同所寫的”Blow-up of the solution of a nonlinear Schrödinger equation system with periodic boundary conditions” [8]。在這兩篇論文中，我們改進了他們的一些結果。我們獲得了包含更多初始值的條件使得解在有限時間內爆炸。在第一篇論文中還有寫到我們可以構造出爆炸解，這個想法來自 F. Merle 教授的類似擾動論點的方法 [15]，我們使用固定點定理去證明修正項的存在性。為了計算方便與更直接的方式我們調整了原始的證明 [8, 17] 以及增加了一些被作者省略的證明細節。我們也修正了一些錯別字和錯誤 [8, 17]。最後選擇第二篇論文中的一個特例，我們給了一個直接的證明關於週期非線性薛丁格系統的爆破解。

In this thesis, we review two papers about the blow-up of solutions for the nonlinear Schrödinger equation and systems with periodic boundary condition. The first paper ”Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition" was written by T. Ogawa and Y. Tsutsumi [17] and the second paper ”Blow-up of the solution of a nonlinear Schrödinger equation system with periodic boundary conditions" was written by F. Ivanauskas and G. Puriu{{s}}kis [8]. In two papers, we improved some of their results. We obtain better conditions that contain more initial data such that the solution blows up in finite time. In the first paper, we can construct a blow-up solution by a method which is some what like the perturbation argument from F. Merle [15]. We use fixed point theorem to prove the existence of the correction term. We modified the original proofs in [8, 17] which make calculations easier and more straight forward. We add some details which are omitted by the authors. Also, we correct some typos and mistakes in [8, 17]. Finally, we give a direct proof of the blow-up solutions of a nonlinear Schrödinger system with periodic boundary conditions which is a special case of the second paper.

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