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.
This project resulted in a number of new mathematical results related to stability of such special solutions of partial differential equations as traveling waves or others more complicated patterns. In particular, we studied various spectral properties of linear approximations of the equations near these solutions. Intellectual Merit. We obtained formulas relating the number of eigenvalues causing instability (called the Morse index) and a geometric quantity (the Maslov index) counting the number of intersections with a given plane of certain path constructed by means of solutions of the respective equation. This is a generalization of the celebrated Morse-Smale Theorem for multidimensional systems on domains with rough boundaries and quite general boundary conditions. In addition, we related the Morse index to the flow of eigenvalues through the origin for a quite general perturbed family of differential operators. On a more applied side, we showed stability of traveling waves for equations arising in chemical kinetics and combustion theory. For systems described by infinitely many parameters we developed a theory of the Evans function, an object generalizing the Wronski determinant familiar from the classical theory of differential equations. To study the Evans function, we involved yet another new in this context tool, the Keldysh Theorem, which allowed us to estimate the number and determine location of the unstable eigenvalues via a robust approximative scheme. Also, we obtained several abstract results in stability theory for general differential equations on infinite dimensional spaces aimed to show how fast a perturbation of a solution dissipates. Broader Impact. This project involved sixteen collaborators from the US, Australia, Germany, Poland, UK, and six PhD students of the Principal Investigator (two of them already graduated and four made an essential progress toward graduation). Based on the results obtained in the course of this project, the PI delivered twenty addresses at various conferences and seminars in the USA and Europe, and organized five special sessions and conferences at leading research centers worldwide. In addition, under PI’s supervision, the graduate students participated in the International Internet Seminar organized by German and Italian mathematicians. The project resulted in fourteen papers and preprints, including publications in the very top mathematical journals worldwide.
