為了調查高能粒子的生成機制和行為，我們選擇研究在Langmuir 孤立波加入速度分布函數。我們首先採用的是模擬同樣具有在位置空間和速度空間訊息的Vlasov方程式。在模擬Vlsov方程式的時候，我們使用的是在相空間維持穩定的splitting scheme。之後我們模擬了線性和非線性的Landau damping,並跟C. Z. Cheng 一九七六年的論文做對照。在典型的速度分布函數上通常都是Maxwellian分布，我們進一步的選擇了使用kappa分布來做考慮，並且觀察波和粒子的交互作用。要了解電子的 Langmuir 波跟離子聲波發生交互作用以後產生的Langmuir孤立波，我們選擇採用離子的Vlasov 方程式。 在初始狀態下，我們所使用的是流體方程式的Zaharov方程式。之後由Zakharov方程式可以得到一個非線性薛丁格方程式，並由這個方程式來決定孤立波的離子密度，(藉由泊松方程式求得)電子密度和電場分布。當電子的分布函數是Maxwellian分佈時，電子很明顯的在高能量的區域被加熱的情形。此外，在研究Langmuir孤立波的時候，我們也把電子的分布函數從Maxwellian分布改為kappa分布。由於kappa分布本來就有著比較多的高能量分布，因此在分布函數上的改變並沒有像Maxwellian分布那麼明顯。

To investigate the behavior of high-energy electrons and their generation mechanism, Langmuir soliton is studied incorporating the velocity distribution function. For the analysis, the Vlasov equation is employed which has the information in both x-space and v-space. In simulating Vlasov equation, we have employed the phase volume conserving splitting scheme. Linear and nonlinear Landau damping is benchmarked with C. Z. Cheng and G. Knorr paper (1976). On top of typical Maxwellian distribution in v-space, kappa dsitribution function is considered, and the wave-particle interaction (Landau damping) is investigated. The ion Vlasov equation is also employed to study Langmuir solitons which arise from the interaction between electron Langmuir waves and ion acoustic waves. For the initial condition, solutions of Zakharov (fluid) equation is employed. The Zakharov system reduces to the nonlinear Schrodinger equation, and this equation provides us with the ion density, electron density (via Poisson equation), and electric field for the soliton profiles. When electron distribution function is Maxwellian distribution, the electrons' high energy tail are heated significantly. Furthermore, the electron distribution is changed from Maxwellian to kappa distribution in the Langmuir soliton study. Due to the kappa distribution's large high energy tail, it demonstrated that the change in the electrons' distribution function is not as prominent as in the Maxwellian case.

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