因為現代光纖通訊系統訊號的光pulse持續時間已壓縮至飛秒(femto-second, 10-15s)以下，如此短的持續時間將使得振幅變的很大，是故傳播時我們必須考慮因光強度所造成的非線性效應(nonlinear effect)，而我的論文就是以Nonlinear Schrödinger equation (NLSE) 作為最基礎的model，然後因應不同情況替換不同的非線性項，觀察各種新的非線性項所帶來的特殊現象。
在介質的分類上，若對入射電磁波的變化立即反應(response)的我們稱此介質為具有瞬時性(instantaneous)特性。不過此種特性只存在於理論上的理想介質，實際的介質都需要一定的反應時間(relaxation time)，為非瞬時性(non-instantaneous)特性，為了描述此特性，我們把最初始的NLSE非線性項替換成具有介質反應時間的非瞬時項，並觀察swinging soliton在此特性下傳播與一般soliton在Kerr medium傳播的不同結果，以及推導non-instantaneous MI在此系統下的不穩定性係數公式。
另外一個主題是關於渦流對的研究，首先我們必須確立特定存在的解型式，本篇論文將以variational method來尋求渦流對(vortex pairs)的解型式。以在nonlocal medium中為例，利用Lagrange equation我們可以找出給定外加激發能量P下與寬度w的關係，滿足此型式的渦流對的解才能滿足原nonlocal NLSE。再來將此型式的vortex pairs當作一背景平面波下的微擾(perturbation)並仿照前述MI的growth rate h2的推導，我們可以依樣畫葫蘆的寫出屬於vortex pairs的h2 公式。

Because the duration of light pulse in modern fiber communication systems is compressed to under the order of femto-second, such a short duration will make large amplitude. We need to consider the nonlinear effect because of the high light intensity when we propagate it in fiber. My thesis is on the basis of Nonlinear Schrödinger equation (NLSE) model, and then we will replace the original nonlinear term to different nonlinear terms with respect to different conditions. By this method we can observe more fantastic phenomenon which is attributed to different new nonlinear term.
By the classification of the media, we call a medium “instantaneous” if it can response instantaneously to the emitting electromagnetic wave. Nevertheless, this property can only exist in the ideal medium by theory. Real media all need some relaxation time to response; therefore real media are “non-instantaneous”. In order to describe the non-instantaneous property, we replace the nonlinear term of the original NLSE to a non-instantaneous term with the parameter of relaxation time τ. We will use this new NLSE to observe the special propagation phenomenon of the swinging soliton and non-instantaneous MI.
However, because NLSE is still a wave equation, the solution needs to be some specific type. My thesis will use variational method to find the specific solution of vortex pairs. At first, we will use the nonlocal medium for example. We can find the relationship of exciting power P and bean width w, and these specific solutions meet the original nonlocal NLSE. It is no different for the case of Kerr medium, although we let these specific solutions of vortex pairs be the perturbation of a background plane wave this time. And then we can derive the formula of the growth rate h2 of vortex pairs in accordance to the similar derivation of MI beforehand.

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