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.

这个项目的主要目标是发展特定的算子理论摄动方法，用于研究应用动力系统中出现的偏微分方程组的行波和其他形式的稳定性问题。计划在Morse和Maslov指数、多维特征值问题(通过Dirichlet-to-Neumann算子的Birichlet-Schwinger扰动行列式)和Evans函数的谱性质等方面给出应用。将算子值亚纯函数的Keldysh型定理应用于关于行波和更复杂的多维模式的线性化的微分算子的谱分析，使用并进一步发展了发展方程的冻结方法。在更广泛的应用方面，非自伴微分算子的谱理论和Evans函数方法，结合关于强连续(但非解析)算子半群的谱性质的抽象结果，将被用来讨论化学动力学和燃烧理论中出现的具体物理模型的行波前锋的非线性稳定性。它包括对复杂系统的性质的研究，这些性质由无限多的参数在时间上演变而成，如它们的稳定性，被理解为在小扰动下保持不变的能力。在这个项目的过程中将使用和进一步发展的主要理论工具是在量子力学和散射理论中使用的无限维矩阵的行列式理论。结合推广微分方程组的Wronski行列式的理论，这将使我们能够计算指示传播波的不稳定程度和其他更复杂的动力学模式的指数。我们将应用这些方法来研究描述固体燃料燃烧的方程和几种化学反应物随时间演变的相互作用。