Mathematical Sciences: Dynamical Systems with Infinitely Many Degrees of Freedom in Mathematical Physics

这个项目将研究具有无穷大的 数学物理中出现的许多自由度 通过诸如Kolmogorov-Arnold-Moser理论的技术， 不变流形定理和李雅普诺夫指数 一是 该项目将侧重于技术之间的关系，如 Kolmogorov-Arnold-Moser理论，以及 收敛的方法来研究局域态， 具有随机或准周期势的薛定谔算子， 并试图利用这些关系来证明 偏微分方程周期轨道的存在性 其次，它将研究如何像中心流形的想法， 定理可以结合一个哈密顿结构来研究 偏微分方程，如Boussinesq方程， 与流体力学有关的其他方程。 最后，它将研究 存在的数量，如李雅普诺夫指数， 无穷维动力系统产生于 力学 更好地理解这些量将使 一个是热力学极限的动力学模拟， 平衡统计力学

This project will study dynamical systems with infinitely many degrees of freedom which arise in mathematical physics through techniques such as the Kolmogorov-Arnold-Moser theory, invariant manifold theorems, and Liapunov exponents. First, the project will focus on the relationship between techniques like the Kolmogorov-Arnold-Moser theory, and the accelerated convergence methods developed to study localized states in Schrodinger operators with random or quasi-periodic potentials, and will attempt to exploit these relationships to prove the existence of periodic orbits in partial differential equations. Second, it will investigate how ideas like center manifold theorem can be combined with a Hamiltonian structure to study partial differential equations like the Boussinesq equation and other equations related to hydrodynamics. Finally, it will study the existence of quantities like the Liapunov exponents for infinite dimensional dynamical systems arising in statistical mechanics. A better understanding of such quantities would give one a dynamical analogue of the thermodynamic limit in equilibrium statistical mechanics.