報告為我們對作者Hans Lindblad.和Avy Soffer.於二零零八年時發表的論文 'A Remark on Long Range Scattering for the Nonlinear Klein-Gordon Equation'並以之為作為底稿並且補充證解。
首先，我們為了觀察解的形式，而先討論線性方程的漸近解，然而求得大致上的形式時我們再進一步去觀察ODE的一階漸近解的形式。進而觀察PDE的一階漸近解形式，最後我們就可以利用前面的部分去觀察PDE高階的漸近解，並且額外說明這些ODE、PDE漸近解他們求解的方式。
我們詳細論述闡明他們的想法及概念，使之更加清晰明白。誠希望本報告能幫助
其他讀者充分且簡易了解本篇論文的內涵。
關鍵詞:
非線性的Klein-Gordon方程; 散射理論。

This paper is based on 'A Remark on Long Range Scattering for the Nonlinear Klein-Gordon Equation',which is published by Hans Lindblad and Avy Soffer in 2008, and we make some additional proofs and interpretations.
First, in order to observe the form of the solution, we discuss the asymptotic solution of the linear equations. When we find the general solution,we observe the form of firsr-order asymptotic solution of ODE,thus we observe the form of firsr-order asymptotic solution of PDE. Finally,we can use the previous parts to observe the form of higherorder asymptotic solution of PDE.In addition, we illustrate the procedure of solving related of ODE and PDE.
We interpret their thinkings and concepts clearly in detail.
We hope this thesis can help readers to understand the meaning of the paper sufficiently and easily.
Key words:
Nonlinear Klein-Gordon Equation; Scattering Theory.

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