Spectral Theory and Applied Dynamical Systems

谱理论和应用动力系统

• 批准号: 1067929
• 负责人: Yuri Latushkin
• 金额: $ 17.79万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1067929&HistoricalAwards=false
• 关键词: Spectral; Theory; Applied; Dynamical; Systems

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.

该项目的主要目标是开发算子理论的特定摄动方法，专门用于研究行波的稳定性问题和应用动力系统中出现的偏微分方程的其他模式。该计划是在莫尔斯指数和马斯洛夫指数、多维特征值问题（通过狄利克雷到诺依曼算子的伯曼-施温格扰动行列式）以及埃文斯函数的谱特性等方向上提供应用。算子值亚纯函数的凯尔迪什型定理将应用于微分算子的谱分析，微分算子表现为行波和更复杂的多维模式的线性化，使用并进一步发展演化方程的冻结方法。在更广泛的应用方面，非自伴微分算子的谱理论和埃文斯函数方法，结合强连续（但非解析）算子半群谱性质的抽象结果，将用于讨论混凝土物理移动前沿的非线性稳定性。化学动力学和燃烧理论中出现的重要模型。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这些特性的研究，这些复杂系统的特性由随时间演变的无限多个参数描述，即稳定性，被理解为在小扰动下保持不变的能力。该项目过程中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行列式理论。结合泛化微分方程朗斯基行列式的理论，这将使我们能够计算指示传播波和其他更复杂的动力学模式的不稳定程度的指数。我们将应用这些方法来研究描述固体燃料燃烧的方程以及几种化学反应物随时间演变的相互作用。