Euler-Poisson方程组是等离子体物理中的经典模型，在航空航天、核能技术、半导体、凝聚态等方面都有重要应用。通过扰动分析方法可从形式上由该模型推导出许多著名的色散偏微分方程，如KdV、KP、NLS、DS等。本项目旨在研究Euler-Poisson方程组及其量子修正模型的逼近理论：1、长波长极限；2、调制逼近。申请人已经在杂志CMP和SIAM上发表了与本项目密切相关的研究成果。在此研究基础上，首先在半空间以及有界区域中建立长波长极限理论，可能出现的边界层以及高阶导数无边界值等问题需要被处理；其次，严格证明冷离子、电子以及量子等模型的NLS逼近结果并在二维空间中考虑DS逼近问题，共振点的出现以及二次项损失导数会在此调制逼近过程中引起本质困难。研究这些色散偏微分方程的极限过程可以有效地观察到等离子体的动态行为，具有重要的物理意义和广阔的应用前景，是同行学者们关心的热点和难点问题之一。

The Euler-Poisson system is a classic model in plasma physics, which is widely used in aerospace, nuclear energy technology, semiconductors, condensed matter research, etc. By using the perturbation analysis method, many well-known dispersive partial differential equations can be derived formally from the Euler-Poisson system, such as the KdV equation, KP equation, NLS equation and DS equation. In this project, we will study the long wavelength limit and the modulation approximation for the Euler-Poisson system and related quantum modified models. We have published some papers closely related to this project in journals such as CMP and SIAM. Based on the previous research, firstly, the long-wavelength limit will be established in half-space and bounded regions. We need to deal with the possible boundary layers and high-order derivatives without boundary values. Secondly, we prove rigorously the NLS approximations for the cold ion Euler-Poisson system, the electron Euler-Poisson system and the quantum Euler-Poisson system respectively, and consider the DS approximation in the two-dimensional space. In the process of modulation approximations, the essential difficulties focus on the appearance of resonances and the loss of the derivatives for the quadratic terms. By studying the approximations to these dispersive partial differential equations, we can observe the dynamic behavior of plasma effectively. It has very important physical significance and broad application prospects and is one of the hot and difficult problems concerned by experts and scholars.