Nonclassical viscous conservation laws arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena, including homoclinic and periodic solutions along with the standard heteroclinic (shock, or front type) solutions. The PI aims to study nonlinear stability of periodic traveling waves in one dimension. Singular perturbation problems form a very challenging class of dynamical systems involving multi-scales. The presence of turning points for some singularly perturbed problems relates to a loss of stability of the dynamical structures. The PI plans to investigate stability issues of this dynamical system. Spectral theory links spectral properties of operators to the geometry and topology of the underlying manifold, so it is critical to have a fine spectral theory. The PI proposes to develop a refinement of the Sacker-Sell spectral theory and moreover investigate the relation of the refinement to other spectral theories.
Qualitative information of differential equations that are commonly used to model interesting phenomena in various areas of science and engineering is highly desirable from the point of view of physical applications, since it is practically impossible to find exact solutions of the differential equations modeling the applications. Stability and spectral theory can provide this information. The questions considered in the project have broad applications, so their resolution holds great promise for widespread use in many areas of science and engineering. Such work would further the understanding of the dynamical properties and provide more insight into the processes where they arise.
