Discovery Projects - Grant ID: DP210101102

哈密顿动力系统的谱理论。非线性哈密顿偏微分方程中稳态、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，可靠地计算特征值是至关重要的，它决定了这种结构的稳定或不稳定。这个项目将开发计算哈密顿算子在多个空间维度上的谱的方法。它将使用强大的几何工具马斯洛夫指数和埃文斯函数。我们将使用这些来同时推进，并将二维欧拉方程和雅可比算子的理论结合在一起。

Spectral Theory of 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.