Research in Applied Dynamical Systems

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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.