薛定谔泊松方程的动力学研究

As a class of Schrödinger equations containing a non-local Poisson potential term, Schrödinger-Poisson equations are the mathematical models arising in electrodynamics and semiconductor theory, and play an important role in celestial mechanics, Plasma Physics etc. The research on dynamical behavior of Schrödinger-Poisson equations is a novel and important topic in the field of differential equations and dynamical systems. Based on comprehensive applications of modern mathematics knowledge (mainly including topological methods and variational methods in nonlinear analysis and bifurcation theory of differential equations), we develop theory of standing wave solutions of Schrödinger-Poisson equations, with emphasis on the existence and multiplicity of sign-changing standing wave solutions, the existence, multiplicity, stability and bifurcation phenomenon of general standing wave solutions, and so on, in order to form a systematic theory and research methods for the dynamics study of Schrödinger-Poisson equations, and to build a solid theoretical foundation and provide the methods to solve problems for researchers in applied fields. The study of dynamics of Schrödinger-Poisson equations not only needs the classic theory of dynamical systems, but also calls for other related knowledge including topology, functional analysis, computational mathematics and so on, cannot only enrich the theory of dynamical systems, but also contribute to novel ideas, new theory and technical methodology in mathematics such as the nonlinear functional analysis. Most importantly, the study of dynamics of Schrödinger-Poisson equations involves the interaction of different branches of mathematics disciplines, and should be of a great importance in both mathematical theory and applications.

薛定谔泊松方程是一类具有非局部泊松项的薛定谔方程，是量子电动力学和半导体理论中抽象出来的数学模型，在天体力学、等离子体物理等学科也有着非常重要的应用。薛定谔泊松方程动力学研究是微分方程和动力系统领域新颖而又重要的研究课题之一。本项目旨在综合运用现代数学知识（主要是非线性分析中的拓扑方法和变分方法以及微分方程分支理论）去发展薛定谔泊松方程驻波解理论，重点研究变号驻波解的存在性与多重性，一般驻波解的存在性、多重性、稳定性和分支等问题，使薛定谔泊松方程动力学研究形成一套比较系统的理论和研究方法，为应用领域的工作者提供可靠的理论依据和解决问题的方法。该项目研究既要用到经典的动力系统理论，又要用到拓扑、泛函分析及计算数学等相关知识，不仅可丰富微分方程与动力系统理论，又可探索数学（尤其是非线性泛函分析）及其交叉应用中的新思想、新理论和新方法，且可使不同数学分支学科之间进行相互交叉与渗透。

薛定谔泊松方程是一类特殊的非局部薛定谔方程，是量子电动力学和半导体理论中抽象出来的数学模型，在天体力学、等离子体物理等学科也有着非常重要的应用。非局部薛定谔方程动力学研究是微分方程和动力系统领域新颖而又重要的研究课题之一。本项目综合运用现代数学知识（主要是非线性分析中的拓扑方法和变分方法以及扰动方法）发展了薛定谔泊松方程以及非局部薛定谔方程驻波解理论，重点研究了具有临界指数增长的薛定谔泊松方程一般驻波解的存在性与多重性，以及解的渐近性质；研究了具有临界指数增长的非局部薛定谔方程变号驻波解的存在性、多重性、分支现象等问题，使得非局部薛定谔方程动力学理论得到补充和完善。该项目研究既用到了经典的微分方程理论知识，又要用到拓扑、泛函分析等相关知识，不仅丰富了微分方程与动力系统理论，又为应用领域的工作者提供可靠的理论依据和解决问题的方法。