非线性薛定谔方程是描述量子力学的基本方程，它在非线性光学、超导、玻色-爱因斯坦凝聚等物理领域有着广泛的应用。非线性薛定谔方程孤立波解（包括行波解和驻波解）的研究有着重要的理论及应用意义。本项目将主要围绕两类重要的非线性薛定谔方程：一类是超导中的非线性Schr?dinger -Poisson方程，另一类是来自于玻色-爱因斯坦凝聚中的Gross-Pitaevskii方程, 对其孤立波解的存在性和稳定性开展深入的数学理论研究。具体研究内容包括：1. 利用变分法和临界点理论研究非线性Schr?dinger-Poisson方程变号驻波解的存在性, 探索无界域上含有非局部项的非线性椭圆型偏微分方程变号解的构造方法. 2. 研究Gross-Pitaevskii 方程行波解的稳定性，探索判别非线性薛定谔方程孤立波解稳定性的新方法。该项目的研究将涉及到非线性泛函分析中诸如变分法及临界点理论的新应用。

Nonlinear Schr?dinger equations (NLS) are the fundamental equations describing quantum mechanics and apply to many physical fields such as nonlinear optics, superconductivity and Bose-Einstein concentration (BEC) and so on. The study on solitary wave solutions, including travelling wave and standing wave solutions, plays an important role in both theory and applications of NLS. In this subject, we will focus on two classes of NLS: one is nonlinear Schr?dinger-Poisson equation from superconductivity, and the other is Gross-Pitaevskii equation from BEC. We will study the existence of nodal type standing wave solution of nonlinear Schr?dinger-Poisson equation by using variational methods and critical point theory.and try to find new methods to construct nodal solutions of nonlinear elliptic equations in unbounded domains with nonlocal terms. We will also study the stability of travelling wave solution of Gross-Pitaevskii equation and give some new ideas to determine the stability of solitary wave solutions of NLS. This subject will involve some new applications of variational methods and critical point theory in nonlinear functional analysis.