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.
