格点系统与离散 Hamilton 系统的基态解

Nonlinear dynamical systems play a very important role in nonlinear science. Lattice systems and discrete Hamiltonian systems are two kinds of very important nonlinear dynamical systems. Homoclinic solutions are one of the important research direction of nonlinear dynamical systems, and it has a wide range of applications in biomathematics, mechanics, chemistry etc. Therefore, the research of homoclinic solutions for lattice systems and discrete Hamiltonian systems has very important theoretical and practical significance. Based on the previous work, we will focus on the existence of ground state solutions (i.e. least energy homoclinic solutions) for lattice systems and discrete Hamiltonian systems by using variational methods. The detailed research contents are as follows: (1) Under more general superquadratic (without (AR) condition), we discuss the existence of ground state solutions for second order lattice systems; (2) Under the assumptions that the perturbation term is neither periodic nor coercive and the potential satisfies asymptotically quadratic condition, we study the existence of ground state solutions for second order discrete Hamiltonian systems; (3) Under non-periodic condition, we deal with the existence of ground state solutions for first order discrete Hamiltonian systems.

非线性动力系统在非线性科学中占有极为重要的地位。格点系统和离散 Hamilton 系统是两类很重要的非线性动力系统。同宿解是非线性动力系统的重要研究方向之一，在生物数学、力学、化学等中具有广泛的应用。因此，格点系统和离散 Hamilton 系统同宿解的研究具有十分重要的理论和现实意义。本项目将在已有工作的基础上，运用变分法对格点系统和离散 Hamilton 系统基态解（也就是最小能量同宿解）的存在性进行研究。具体内容如下：（1）在更一般的超二次条件（没有（AR）条件）下，讨论二阶格点系统基态解的存在性；（2）在扰动项是既非周期又非强制的以及位势满足渐近二次条件假设下，讨论二阶离散 Hamilton 系统基态解的存在性；（3）在非周期条件下，讨论一阶离散离散 Hamilton 系统基态解的存在性。

非线性动力系统在非线性科学中占有极为重要的地位。反应扩散系统和离散 Hamilton 系统是两类很重要的非线性动力系统，同宿解和精确解是非线性动力系统的重要研究方向之一，在生物数学、力学、物理等中具有广泛的应用。因此，反应扩散系统和离散 Hamilton 系统相关解的问题研究具有十分重要的理论和现实意义。本项目将在已有工作的基础上，运用李群方法对反应扩散系统的精确解和守恒律进行研究，同时运用变分法对离散 Hamilton 系统同宿解的存在性进行研究。具体内容如下：（1）讨论了一类Broer-Kaup方程（特殊的反应扩散系统，可以看作相应格点系统的连续系统）的精确解和守恒律；（2）在更一般的渐近二次和超二次条件下，讨论了一类二阶离散Hamilton系统同宿轨的多重性，建立了一些新的存在性准则, 改进了已有文献的结果。