一维格点上FPU型系统与离散非线性薛定谔方程的可解性研究

项目名称： 一维格点上FPU型系统与离散非线性薛定谔方程的可解性研究

项目编号： No.11501280

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 孙吉江

作者单位： 南昌大学

项目金额： 18万元

中文摘要： 本项目研究一维格点上带有相互作用势的FPU型格点系统非平凡孤立波及周期运动的存在性以及离散非线性薛定谔方程同宿轨的存在性问题，丰富完善已有结果。主要内容包括：1)对具有更一般超二次增长的势函数的FPU型系统，利用广义Nehari流形方法研究其基态周期解的存在性；2)利用极小极大方法、广义弱环绕定理及周期逼近技巧等工具研究具有渐近二次增长的势函数的FPU系统周期运动的存在性，回答部分公开问题并完善存在性结果；3)利用无穷维同调理论、集中紧性原理及延拓方法等工具研究正定和强不定情形下格点系统孤立波多包解的存在性；4)利用集中紧性原理及广义弱环绕定理等工具研究更一般非线性增长条件下离散薛定谔方程同宿解的存在性与多重性。格点系统可以看作某些非线性波方程关于空间变量的离散化，和非线性波方程一样具有很强的物理背景。本项目中问题的提出和解决，将有助于深化格点系统的研究并拓展该理论在相关领域中的应用。

中文关键词： 格点动力系统；周期解；同宿轨；变分法；Morse理论

英文摘要： We mainly study the existence of solitary waves and periodic motions of FPU type systems with nearest neighbor interaction potentials and homoclinic orbits of discrete nonlinear Schrödinger equations in one dimensional lattices, enrich and consummate the well known results. Our main contents are as follows: 1) Under more general superquadratic growth of potential functions, we study the existence of ground state periodic solutions for the FPU type systems by using generalized Nehari manifold approach; 2) We study the existence of periodic motions for the FPU type systems with asymptotically quadratic potentials by using minimax method in combination with generalized weak linking theorem, periodic approximation technique and so on, answer some open problems and consummate the existence results; 3) We study the existence of multibump type solitary waves for for the both definite and strongly indefinite lattice systems by using infinite dimensional homology theory in combination with concentration-compactness principle and extension method; 4) we study the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations with more general conditions by using concentration-compactness principle and generalized weak linking theorem. Lattice systmes can be viewed as the nonlinear wave equations about space variables discretization, and also have strongly physical background. By proposing and solving the problems in the project, it will help to deepen studies of lattice dynamical systems, and it will extend appliciations of this theory in related fields.

英文关键词： Lattice dynamical systems;Periodic solutions;Homoclinic orbits;Variational method;Morse theory