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.

该项目的主要目的是开发用于量身定制的操作者理论的特定扰动方法，该方法是针对行驶波的稳定性问题和其他模式的稳定性问题，用于在应用动态系统中产生的部分微分方程。该计划是在诸如Morse和Maslov指数，多维特征值问题（通过Dirichlet到Neumann Operators的Birman-Schwinger扰动决定因素以及Evans功能的光谱特性的指示下）提供应用。 Keldysh的类型定理用于算子有价值的Meromororphic函数将应用于差分运算符的频谱分析，这些分析看起来像是有关行驶波和更复杂的多维模式的线性化，并进一步开发了用于进化方程的冷冻方法。在更应用的方面，非频谱的光谱理论和Evans功能方法与强烈连续（但不是分析性）操作员半群的光谱特性的抽象结果相结合，将用于讨论流行前线的非线性稳定性，以实质性地进行流行前线的稳定性。在化学动力学和燃烧理论中产生的重要模型。该提案的主题位于应用和纯数学领域的交集。它包括对许多参数描述的复杂系统的这种特性的研究，这些参数会随着时间的流逝而演变为稳定性，被理解为在小扰动下保持保留的能力。在该项目过程中将使用并进一步开发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决定因素。结合介绍了微分方程的wronski决定因素的理论，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性程度。我们将把这些方法应用于描述固体燃料燃烧以及几种化学反应物的时间相互作用的方程研究。