我們仔細研讀了S.A. El-Wakil及 M.A. Abdou 教授發表的論文"New exact travelling wave solutions of two nonlinear physical models" 還有王明亮、王躍明、張金良教授共同發表的論文"The periodic wave solutions for two systems of nonlinear wave equations" 然後探討文章內所使用的方法並加以應用。
在這篇論文中，我們介紹了幾種不同的偏微分方程，然後再以一種改進的tanh函數方法與計算機符號計算去找這些非線性系統的精確旅行波解，包括Zakharov 方程組、KdV 方程組...等，以及Quantum Zakharov 方程組的近似解。 然後代入一些參數，再將其解的圖形用Mathematica畫出，這些告訴我們方程式有不同類型的解，他們有其代表的物理意義，過程中我們會認識更多偏微分方程之間的共通性，在推導的過程上也會詳細的描述和介紹。
該方法的主要思想是充分利用Riccati方程的家族解，去獲得了精確的解，包括新的類孤子解，三角函數解和有理解。最後在補充了一些作者沒有寫出來的解，並修正維基百科上的錯誤，讓結果更加完整。

We mainly study two papers "New exact travelling wave solutions of two nonlinear physical models" written by S.A. El-Wakil and M.A. Abdou and "The periodic wave solutions for two systems of nonlinear wave equations" written by Wang Ming-Liang, Wang Yue-Ming, and Zhang Jin-Liang , then we explore the methods used in the article and use them to find the exact traveling wave solutions of some partial differential equations.
In this paper,we introduced several different partial differential equations, and use an improved tanh function method and computer symbolic calculation to find exact travelling wave solutions for some nonlinear systems, including Zakharov equations, KdV equations ... etc, and the approximate solution of Quantum Zakharov equations. Then substitute some proper parameters, and then use the Mathematica to draw the solution graphs. These tell us that the equations have different types of solutions and they have their own physical meanings. In the process, we will learn more about the commonalities between partial differential equations, and we will describe and introduce them in detail in the derivation process.
The main idea of this method is to take full advantage of the family of Riccati equation which has more solutions. The exact solutions obtained are including new soliton-like solutions, trigonometric function solutions, and rational solutions. Finally, I added some solutions that the author did not presented, and corrected some errors on Wikipedia to make the results more complete.

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