Spectral Theory of Hamiltonian Dynamical Systems

哈密​​顿动力系统的谱理论

• 批准号: DP210101102
• 负责人: Prof Holger Dullin
• 金额: $ 21.79万
• 依托单位: The University of Sydney
• 依托单位国家: 澳大利亚
• 项目类别: Discovery Projects
• 财政年份: 2022
• 资助国家: 澳大利亚
• 起止时间: 2022-01-01 至 2024-12-31
• 项目状态: 已结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP210101102
• 关键词: Spectral; Theory; Hamiltonian; Dynamical; Systems

Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial dimension. It will use the powerful geometric tools of the Maslov index and the Evans function. We will use these to simultaneously advance, and bring together the theories of the two dimensional Euler equations and Jacobi operators.

非线性哈密顿偏微分方程中的稳态、行波、周期波和其他相干结构的稳定性理论是现代动力系统的基石。特别是可靠地计算特征值至关重要，特征值决定了此类结构的稳定性或不稳定性。该项目将开发计算多个空间维度的哈密顿算子谱的方法。它将使用马斯洛夫指数和埃文斯函数等强大的几何工具。我们将利用这些同时推进，并将二维欧拉方程和雅可比算子的理论结合起来。