Spectral Theory and Applied Dynamical Systems

本项目的主要目标是发展特定的微扰算子理论方法，以研究应用动力系统中出现的行波和其他模式的偏微分方程的稳定性问题。该计划将在莫尔斯和马斯洛夫指数、多维特征值问题（通过Dirichlet-to-Neumann算子的Birman-Schwinger摄动行列式）和埃文斯函数的谱性质等方向上给出应用。算子值亚纯函数的Keldysh型定理将应用于行波和更复杂的多维模式线性化的微分算子的谱分析，使用并进一步发展演化方程的冻结方法。在更多的应用方面，非自伴随微分算子的谱理论和Evans函数方法，结合强连续（但不是解析）算子半群谱性质的抽象结果，将用于讨论化学动力学和燃烧理论中产生的具体物理重要模型的行进锋面的非线性稳定性。本提案的主题位于应用数学和纯数学的几个领域的交叉点。它包括对复杂系统的特性的研究，这些特性是由无限多个随时间演变的参数描述的，比如它们的稳定性，被理解为在小扰动下保持不变的能力。本项目将使用和进一步发展的主要理论工具是量子力学和散射理论中使用的无限维矩阵的行列式理论。结合微分方程的朗斯基行列式的推广理论，这将使我们能够计算指示传播波和其他更复杂的动力模式的不稳定程度的指标。我们将把这些方法应用于描述固体燃料燃烧的方程和几种化学反应物随时间变化的相互作用的研究。

The main objective of this project is to develop specific perturbation methods of operator theory tailored to the study of stability issues of traveling waves and other patterns for partial differential equations arising in applied dynamical systems. The plan is to give applications in such directions as Morse and Maslov indices, multidimensional eigenvalue problems (via the Birman-Schwinger perturbation determinants for the Dirichlet-to-Neumann operators), and the spectral properties of the Evans function. Keldysh' type theorems for operator valued meromorphic functions will be applied to the spectral analysis of the differential operators that appear as linearizations about traveling waves and more complicated multidimensional patterns, using and further developing the freezing method for evolution equations. On the more applied side, the spectral theory of nonselfadjoint differential operators and the Evans function approach, combined with abstract results on spectral properties of strongly continuous (but not analytic) operator semigroups, will be used to discuss nonlinear stability of traveling fronts for concrete physically important models arising in chemical kinetics and combustion theory.The topic of this proposal is situated at the intersection of several areas of applied and pure mathematics. It includes the study of such properties of complex systems described by infinitely many parameters evolving in time as their stability, understood as ability to stay preserved under small perturbations. The main theoretical instrument that will be used and further developed in the course of this project is the theory of determinants of infinite dimensional matrices utilized in quantum mechanics and scattering theory. Combined with the theory generalizing Wronski determinants of differential equations, this will allow us to compute indices indicating the degree of instability of propagating waves and other more complicated dynamical patterns. We will apply these methods to the study of equations describing combustion of solid fuels and of the evolving in time interaction of several chemical reactants.