Spectral Theory and Applied Dynamical Systems

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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.