本项目主要研究Gross-Pitaevskii方程和Ermakov系统解的存在性和分类，及与它们相关的若干问题，其中包括Kadomtsev-Petviashvili方程及Ginzburg-Landau方程解的分析等。这些问题都具有很强的物理背景，与凝聚态物理，超导理论，浅水波理论，非线性光学等有关。另一方面，它们与几何及可积系统理论也具有紧密联系。在本项目中，我们将利用Lyapunov-Schmidt约化，临界点理论，爆破分析等方法，结合可积系统及哈密顿系统等方面的理论和方法，研究上述方程的解空间的结构。本项目的顺利实施将深化我们对GP，Ermakov，Ginzburg-Landau，KP等几大类方程的理解。同时，关于KP方程的研究在四阶椭圆方程的理论中也具有重要意义。

In this project, we will mainly study existence and classification of solutions for Gross-Pitaevskii and Ermakov system, and some closely related equations, such as Kadomtsev-Petviashvili equation and Ginzburg-Landau equation. These equations are of physical importance, and appear in various physical contexts, such as Bose-Einstein condensate, superconductivity theory, shallow water wave, nonlinear optics. On the mathematical side, they are connected to geometry and integrable system theory. Using Lyapunov-Schmidt reduction, critical point theory, blow up analysis, theory and method form integrable system and Hamiltonian system, the structure of solution sets will be studied, which will then enable us to understand GP, Ermakov, KP and Ginzburg-Landau equations better. It can be expected that our investigation of KP equation will also be important in the theory of fourth order nonlinear elliptic equations.