Eigenvalue and Stability Problems in Applied Mathematics

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The research that will be supported by this award addresses a number of topics in applied mathematics that have the common theme of eigenvalue problems. Eigenvalue problems arise in a number of contexts in various branches of science, in particular in questions related to the stability of a physical system. Specifically, in many equations arising in mathematical physics there may exist valid exact solutions which are unstable, in the sense that nearby solutions rapidly diverge from this exact solutions as time progresses. Such solutions are often difficult or impossible to realize in an experiment, since the initial conditions must be chosen very precisely in order for the solutions to remain observable for any appreciable length of time. Put differently, physically interesting solutions typically are the stable ones. and what is not. Unfortunately it is often very difficult to determine whether a particular solution is stable or not. The problems that will be studied under this award range from water waves and wave propagation in photonic materials to models in physiology. Specifically, problems related to photonic crystals with defects, standing waves for nonlinear Schroedinger equations, vortex crystals for Bose-Einstein condensates, scattering problems related to the Sine-Gordon equations, and a problem from neurophysiology (oculomotor integrator of the brain) will studied. In some of these projects, geometric methods for establishing the instability of solutions will be developed. These methods are very general, being based on geometric considerations rather than details of the equation in question. Thus they are potentially applicable to problems in many disciplines within science. Other techniques such as asymptotics and topological arguments will also be employed. Stability is a fundamental property of many physical systems. For example, a pocket watch hanging from a chain is a stable system - it will eventually return to its rest state when perturbed. A pencil that is balancing on its tip on a table is unstable - a very small perturbation will cause it to fall over. For many engineered physical systems, it is however not clear if it is stable or not. Examples occur in the design of novel optical materials, where the stability of a traveling light pulse is of interest. In other situations, e.g. in physiology, instability is related to malfunction, and it is of interest to explore mechanisms that lead to instability. For example, the oculomotor integrator, which is part of the brain subsystem that moves the eyes, is known to function properly as long as a mathematical model satisfies a suitable condition (a single dominant eigenvalue is near the origin). The system becomes unstable if there are eigenvalues in locations where they not are supposed to be, resulting in a condition known as congenital nystagmus that effects one in about 2000 people. The research in this project is primarily concerned with questions of stability in the broadest sense. It will provide tools that are applicable to problems ranging from photonics to physiology.