Research in operator theory and applied dynamical systems

该项目的主要目标是深入研究Evans函数的算子理论性质，Evans函数是当代行波和波型稳定性理论的主要工具之一。 我们计划开发一个形式主义的埃文斯函数在无限维设置的抽象类方程的有界扰动的对角一致稳定的双半群，包括（在应用侧）线性抛物问题的无限圆柱，和许多其他。本文利用Krein的谱位移函数导出了抽象微分算子的Fredholm指标的一个新公式。 我们将探索埃文斯函数和修改后的Fredholm行列式的Birman-Schwinger型积分算子之间的强大和令人惊讶的联系。这包括一个新的抽象公式的衍生物的修改Fredholm行列式在本征值，其具体实现的衍生物的Jost和埃文斯功能用于检测不稳定的本征值的线性化沿着脉冲。 特别地，我们将对一维固体中燃烧波理论中的一个重要应用问题进行Evans函数谱分析，这个建议的主题是复杂动力系统的应用理论和算子谱理论的相互作用，算子谱理论是包含无穷多个参数的方程的数学理论。我们将研究行波，脉冲，前沿和其他模式可能描述燃烧，波传播和许多其他自然现象。我们的主要应用目标是开发工具来理解模式是否稳定，也就是说，它们的结构是否在小扰动下保持不变。在这一领域带来一些想法从量子力学，我们将推广和利用系统包含无穷多个参数的概念的埃文斯函数，行列式类似于朗斯基在微分方程理论。我们的主要理论贡献将是在统一定理的证明，使用埃文斯函数和描述的方程线性化的上述模式通过渐近性质的微分方程包含无穷多个参数，并与稳定性的模式的可解性。

The main objective of this project is to give an in-depth study of the operator-theoretic nature of the Evans function, one of the principal tools in contemporary stability theory of travelling waves and wave patterns. We plan to develop a formalism for the Evans function in the infinite dimensional setting for an abstract class of equations obtained as bounded perturbations of diagonal uniformly stable bi-semigroups, including (on the applied side) linearized parabolic problems in infinite cylinders, and many others. We will derive a new formula for the Fredholm index of abstract differential operators in terms of Krein's spectral shift function. We will explore powerful and surprising connections between Evans functions and the modified Fredholm determinants of the Birman-Schwinger-type integral operators. This includes a new abstract formula for the derivative of the modified Fredholm determinant at an eigenvalue, and its concrete realization for the derivatives of the Jost and Evans functions used to detect unstable eigenvalues of linearizations along pulses. In particular, we will carry out the Evans function spectral analysis of an important applied problem in the theory of combustion waves in one-dimensional solids.The topic of this proposal is an interplay of applied theory of complex dynamical systems and operator spectral theory, the mathematical theory of equations containing infinitely many parameters. We will study travelling waves, pulses, fronts, and other patterns potentially describing combustion, wave propagation, and many other natural phenomena. Our main applied goal is to develop tools for understanding whether the patterns are stable, that is, whether their structure is being preserved under small perturbations. Bringing in this area some ideas from quantum mechanics, we will generalize and utilize for systems containing infinitely many parameters the concept of the Evans function, a determinant similar to the Wronskian in the theory of differential equations. Our main theoretical contribution will be in the proof of unifying theorems that use the Evans function and describe solvability properties of equations linearized about the above-mentioned patterns via asymptotic properties of differential equations containing infinitely many parameters, and related to stability of the patterns.