這篇論文在探討孤立波在非線性介電質內傳播的理論與數值模擬，內容包含了基礎電漿物理與光纖的訊號傳播。電漿和光纖都是屬於非線性介電質，因此我們以Zakharov方程式和非線性薛丁格方程式為基礎來研究孤立波的行為。在這我們使用有限差分法來模擬非線性方程式。
在電漿中，Langmuir孤立波是由有質動力產生的，且藉由改變背景密度，孤立波可被加速。若孤立波的群速度小於聲速，則孤立波可維持原狀傳播，且運動積分不變。但若我們使孤立波加速，則孤立波會放射聲波而導致孤立波崩潰。我們也探討雙孤立波的行為，並發現即使沒有給予初始速度，兩個孤立波也會彼此互相靠近。假設我們擁有穩態密度這個特殊條件，也證明能將Zakharov系統簡化為非線性薛丁格方程式。
在非線性薛丁格方程式中，我們謹慎地檢視了在均勻和非均勻的介質內孤立波的行為。在此我們使用折射率的不同來產生非均勻介質的效果。

This thesis research is about theoretical and numerical studies of solitary wave propagation in nonlinear dielectric media. The research is related to both fundamental plasma physics and signal processing by optical fibers.
To investigate the soliton behavior in plasma and in optical fibers as one of the examples of the dielectric medium, we start from Zakharov equations and nonlinear Schrödinger equation. Finite difference methods are employed to simulate the nonlinear equations numerically. In plasma, ponderomotive force produces the Langmuir soliton. By changing background density, the soliton will be accelerated.
If the solitons’ group velocity is smaller than that of sound wave, the soliton can propagate with a constant shape retaining integral of motion constant. However, if a significant acceleration is given, solitons can collapse and they can be distorted by emitting the sound wave. Furthermore, by providing two solitons, they can merge without giving them any initial velocity. In special cases, assuming steady state density, we demonstrate the Zakharov system can reduce to that of nonlinear Schrödinger equation.
For nonlinear Schro ̈dinger equation, we examine the solitons propagation under uniform and nonuniform media, which is releated to variation of the index of refraction.

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