This project concerns systems with multiple length and time scales, with the goals of analyzing recent experiments, of improving computational methods, and of establishing new mathematical theory for such systems. First, in the area of chemical patterns with multiple length scales, the recently-discovered phenomena of self-replicating spots and pulses has posed new challenges for modeling and for stability analysis of solutions of partial differential equations. The investigator and collaborators build on their analysis of the dynamics, time scales, and mechanisms responsible for self-replication to study the underlying bifurcation hierarchies that organize the self-replication regime, to further examine the zero-pole cancellation phenomenon in the nonlocal eigenvalue problem stability analysis, to extend the renormalization group technique to establish the fully-nonlinear stability of pulses, and to develop extensions to systems with more than two length scales. Second, in the increasingly-important area of reduction methods for large systems of chemical reactions with multiple time scales, the validity and accuracy of certain methods are analyzed, with special focus on the computational singular perturbation method of Lam and Goussis. Third, the investigator analyzes the Oya-Vallochi model of subsurface bioremediation. Bioremediation is a process in which microorganisms, in the presence of electron acceptors, degrade environmentally-harmful organic compounds. The investigator studies traveling waves of biomass activity and advection versus dispersion. Fourth, he conducts fundamental studies of nonspherical deformations of gas bubbles in Newtonian fluids. Finally, a challenging open problem concerning the existence of self-similar, blow-up solutions of the nonlinear Schroedinger equation in spatial dimensions between two and four is attempted. This project concerns mathematics for problems of significant current interest in biology, chemistry, engineering, and physics, which exhibit both fast and slow dynamical processes. First, with collaborators and a doctoral student, the investigator analyzes computational methods used to simulate large, complex systems of reactions in biochemistry, combustion, and air pollution engineering. These processes, such as the production of certain proteins, the burning of natural gas, and the formation of nitrous oxides in the atmosphere, typically involve a few hundred species, each of which participates in several reactions, with the reaction times ranging from nanoseconds to milliseconds, even to minutes. Methods that reduce the system complexity, while retaining a desired accuracy, are critical for modeling these processes. The investigator aims to show that there is a highly accurate method that can be used to improve the accuracy of other widely-used methods, which are embedded in major computer codes. Second, the investigator and a doctoral student study mathematical models of bioremediation, in which microorganisms are used to degrade environmentally-harmful organic compounds. Mathematics provides an advantageous approach to determine important quantities, such as the wave speed with which the biologically-active zone propagates through a wet soil column and how this speed depends on the many physical parameters. Third, fundamental research is conducted on the dynamics of gas bubbles in water. Deformations of spherical bubbles lead to oscillations on time scales much shorter than that on which the spherical mode itself oscillates, and the main goal is to model the nonlinear transfer of energy between the spherical and nonspherical modes that can lead to bubble cavitation and the attendant production of underwater sound by turbine blades, for example. Finally, the investigator develops further theory for self-replicating chemical patterns and for a prototypical equation that governs nonlinear wave propagation.
