Research in Applied Dynamical Systems

这个项目的重点是进一步发展被称为马斯洛夫指数的数学方法。这种理论仪器是一种几何计数，它描述了行波的不稳定性程度，以及在偏微分方程的解中发现的其他更复杂的模式。它借鉴了一系列广泛的工具，包括谱和摄动理论、辛几何和哈密顿动力学。虽然马斯洛夫指数最初是在数学的一个抽象领域引入的，但当前项目的主要目标之一是证明它在许多未解决的和更多应用问题中的基本重要性。这种方法可以用于研究描述固体燃料燃烧和几种化学反应物随时间变化的相互作用的方程，以及描述理想不可压缩流体的正则化方程的稳定性，这在三个多世纪以来一直是研究的挑战。多维反应扩散和其他偏微分方程的非线性波和其他模式的稳定性和不稳定性，以及这些特殊解附近的动力学研究，是当代应用动力系统、偏微分方程、非自伴随算子的谱理论和无限维辛几何十字路口的基石问题。本项目旨在发展和应用多维微分算子谱理论的新方法，并解决应用燃烧理论模型的非线性稳定性问题，以及反应扩散方程的一般梯度系统，以及许多其他问题。该项目建立在最近在这些领域通过马斯洛夫指数计算莫尔斯指数取得的理论突破的基础上。莫尔斯指数计算模式线性化的不稳定特征值的个数，而马斯洛夫指数是辛几何中的不变量，用于计算无限维拉格朗日平面空间中路径相交的有符号数。首席研究员和他的同事们的目标是进一步发展马斯洛夫指数方法，通过证明新的hadamard型公式的特征值的导数和新的谱流公式的微分算子族的线性化关于行波的偏微分方程。马斯洛夫指数计算被用于应用动力系统中的几个开放问题：图灵模式的梯度猜想、非线性谱的计算和非线性薛定谔方程的周期问题。该项目的一个重要部分是研究在固体燃料燃烧理论和放热和吸热化学反应建模中出现的反应扩散系统的平面前沿的多维非线性稳定性。

The focus of this project is to further develop the mathematical approach known as the Maslov index. This theoretical instrument is a geometric count that describes the degree of instability of traveling waves and other more complicated patterns found in solutions of partial differential equations. It draws upon a broad array of tools, including spectral and perturbation theories, symplectic geometry, and Hamiltonian dynamics. While the Maslov index was originally introduced in an abstract area of mathematics, one of the major objectives of the current project is to demonstrate its fundamental importance in many unresolved and more applied problems. This approach can be used to study equations describing the combustion of solid fuels and the interaction of several chemical reactants evolving in time, as well as the stability of a regularized version of equations describing ideal incompressible fluids that has continued to challenge researches for more than three centuries.Stability and instability of nonlinear waves and other patterns for multidimensional reaction diffusion and other partial differential equations, and the study of dynamics near these special solutions, are cornerstone issues at the crossroads of contemporary applied dynamical systems, partial differential equations, spectral theory of non-selfadjoint operators, and infinite dimensional symplectic geometry. The aim of this project is to develop and apply new methods in spectral theory of multidimensional differential operators, and address nonlinear stability for models in applied combustion theory, as well as general gradient systems of reaction diffusion equations, and many others. The project is built on a theoretical breakthrough recently achieved in these areas in computing the Morse index via the Maslov index. The Morse index counts the number of unstable eigenvalues of the linearization about the pattern, while the Maslov index is an invariant from symplectic geometry that counts the signed number of intersections of paths in the space of infinite dimensional Lagrangian planes. The principal investigator and his colleagues aim to to further develop the Maslov index approach by proving new Hadamard-type formulas for the derivatives of the eigenvalues and new spectral flow formulas for families of the differential operators obtained by linearizing partial differential equations about the traveling waves. The Maslov index calculations are being utilized in several open problems in applied dynamical systems: the gradient conjecture on Turing's patterns, the computation of spectra of nonlinear pencils, and the periodic problems for nonlinear Schrodinger equations. An important part of the project is the study of the multidimensional nonlinear stability of planar fronts for reaction diffusion systems arising in combustion theory of solid fuels and in modeling of exothermic and endothermic chemical reactions.