我們仔細研讀了M. Ohta 教授的論文”Stability of solitary waves for the Zakharov equations in one space dimension” 與M. Maeda 教授的論文“Stability of ground states of NLS with fourth order dispersion”並且闡述了相關細節在本篇報告裡。在這兩篇論文中，都是討論方程的孤立波解，也可稱作駐波解，與它本身的穩定性，在能量最低的時候，解是穩定的。而我們先針對Zakharov 方程去作探討，利用Zakharov 方程與薛丁格方程的相似之處，用變分法的工具來完成證明。在M. Maeda 教授這篇論文中，我們
針對擁有四次微分項的薛丁格方程進行探討，藉著這極小的四次微項，來讓二維薛丁格方程的最低能量解達到穩定。而我們除了詳讀這兩篇論文之外，也補充了作者們省略的證明細節，以及修正一些打印等方面的小錯誤。

We study the work of professor M. Ohta, ”Stability of solitary waves for the Zakharov equations in one space dimension”, and the work of professor M. Maeda, ”Stability of ground states of NLS with fourth order dispersion”, then elaborate details in this article.
In both papers, we discuss the solitary wave solutions of the equations, also known as standing wave solution, and the stability. With the lowest energy, the solution is stable. We first discuss the Zakharov equation and use the similarities between the Zakharov equation and the Schrödinger equation to complete the proof, using variational approach. In professor M. Maeda’s paper, we discuss the Schrödinger equation with a fourth-order
derivative term. With small parameter in fourth-order term, the minimum energy solution of the two dimensional Schrödinger equation is stable. In addition to reading these two papers, we also added some details that the authors omitted, as well as some errors in printing.

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