二维格系统的行波解与分支

Lattice differential equations(LDEs) can be discretizations of partial differential equations, and models involving LDEs can be encountered in a wide variety of scientific disciplines, including chemical reaction theory, image processing, material science and biology etc. We first study the existence and multiplicity of wave trains on a two-dimensional Frenkel-Kontorova(FK) lattice with Hamiltonian, by means of the representation theory of Lie groups and center manifold theory which are completely different with the methods of upper-lower solutions and monotone iteration used in LDEs with monotonicity or quasi-monotonicity. In addition, the existence of other travelling waves like heteroclinic orbit are also presented. Then we investigate the two-dimensional damped and forced lattices with local interactions and non-local interactions, respectively. Due to the absence of Hamiltonian, we use the variational method and topological method rather than the representation theory of Lie groups to analyze the existence and uniqueness of periodic travelling waves. The bifurcation of travelling waves in resonance condition is derived using the methods of Lyapunov-Schmidt reduction and functional analysis. In particular, we show the impact of the direction of propagation and forcing and obtain much richer dynamics than in one-dimensional LDEs. This project covers the thought of algebra、geometry、 functional analysis etc. The results obtained in it are novel and provide important theoretical foundation for some application fields.

格微分方程可看作偏微分方程的空间离散化，同时在化学反应，图像处理，材料学和生物学等领域有着广泛的应用。一方面我们考察了一类具有哈密尔顿性的二维FK格系统，与运用上下解的单调迭代法讨论单调或拟单调的格系统不同，本项目将运用李群表示论和中心流形约化方法分析系统周期波列解的存在性和多重性，以及其他行波解如异宿轨的存在性，发展二维格系统不变流形理论。另一方面探讨带有阻尼和受迫项的具有局部相互作用和非局部相互作用的二维格系统。由于哈密尔顿性的缺失，不能运用李群论的方法，而采用非线性的变分法和拓扑法等分析系统周期行波解的存在性和唯一性。特别地，我们将分析行波方向和外力项对系统的影响，综合运用Lyapunov-Schmidt简约和泛函分析等方法分析共振情形下行波解的分支，得到比一维格系统更丰富的动力学性质。本项目涉及代数、几何、泛函分析等多方面的思想，所得到的结果新颖，并为应用研究提供了重要的理论指导。

格微分系统是由定义在具体几何结构上的常微分方程，偏微分方程或差分方程组成的无穷维系统，主要应用于化学反应，图像处理，材料科学以及生物学等领域。本项目主要研究了一类具有阻尼项和周期外力项的一般二维Frenkel-Kontorova格系统的周期运动波解的存在性和多重性，运用变分法讨论了几种周期解的存在性和唯一性，并结合Lyapunov-Schmidt 约化方法和扰动理论得到了共振情形下解的分支和多重性。 同时，对状态依赖时滞微分方程的动力学性质进行了研究，分析了具有状态依赖时滞Nicholson飞蝇模型解的相关性质以及慢振荡周期解的存在性； 接着，利用Browder不动点定理探讨了一类具有状态依赖时滞的二阶非线性微分方程的慢振荡周期解的存在性。该项目的研究不仅可以将泛函微分方程理论与格微分方程结合起来，而且可以发展高维格系统的动力学理论，具有重要的理论和实际应用价值。