梁方程拟周期驻波理论

This project is based on nonlinear beam equations and is intended to focus on the quasi-periodic standing wave problem in space. Standing waves are widely used in mathematics, physics, engineering, materials science, quantum mechanics and other important scientific and technological fields. The related problems have been one of the most important research topics in the field of mathematical physics and applied mathematics and have attracted extensive attention of many mathematicians and physicists for a long time. At present, the quasi-periodic problems in space, including dynamical properties and Cauchy problems, are unclear. The ones are still a new direction that people are actively exploring.. The purpose of this project will be to develop the Lyapunov-Schmidt reduction together with the Nash-Moser iteration. By virtue of these methods, we will study the existence of standing waves with a single frequency and quasi-periodic structures in space for nonlinear beam equations. Furthermore, we will explore the existence of standing waves with non-single frequencies and quasi-periodic structures in space. Since the quasi-periodic structures in space reflect its anisotropic nature, the two kinds of standing waves studied in this project can better reflect the physical properties of practical problems. Through these studies, it is helpful to understand the dynamic mechanism for beam equations, and to provide theoretical basis and research methods for analyzing and dealing with practical problems.

本项目以非线性梁方程为模型，拟围绕其空间拟周期驻波问题展开。驻波广泛应用于数学、物理、工程、材料学、量子力学等重要科技领域，相关问题是当今数学物理和应用数学领域中高度关注的研究课题之一，长期以来备受众多数学家、物理学家的广泛关注。目前，包括动力学性质、Cauchy问题等在内的空间拟周期问题尚不清晰，仍是人们积极探索的新方向。.本项目旨在发展Lyapunov-Schmidt约化和Nash-Moser迭代方法，研究非线性梁方程空间拟周期的单一频率驻波解的存在性，并进一步探讨其空间拟周期的非单一频率驻波解。空间的拟周期结构反映了空间具有各向异性的性质，本项目研究的这两类驻波更能反映实际问题的物理属性。通过这些研究，有助于理解梁方程的动力学机制，为分析和解决实际问题提供理论依据和研究方法。

本项目发展了Nash-Moser迭代方法，隐函数定理，压缩映像原理建立了梁方程、波方程、Kirchhoff方程的周期解或拟周期行波解的存在性。另外，发展了位势理论和谱方法给出了弹性波方程和声弹耦合方程散射场低频部分的共振模态展开形式。通过这些研究，进一步加强对偏微分方程动力学机制的理解，为分析和解决实际问题提供理论依据和研究方法。