Stuart P. Hastings - US grants

Work will be done on problems involving nonlinear differential equations. The work will focus on three areas. In the first, investigations will be carried out on the relation between Painleve transcendents and certain linear integral equations arising in inverse scattering theory. Certain ordinary differential equations in the complex plane have only poles for singularities. These equations, or transcendents, are related to nonlinear partial differential equations solvable by the method of inverse scattering. At this time only selected examples illustrate the connection but it is clear that an entire class of equations can be treated. A systematic study of the entire relationship will be undertaken. A second line of research deals with the failure or blow-up of solutions of nonlinear diffusion equations. Blow-up of solutions depends heavily on the nature of initial conditions (as well as the nonlinearity term in the equation). Since one is often dealing with functions of several space variables, blow-up can occur at different points at different times. This leads to the main conjecture of the project: blow-up must be confined to a set of space variables of negligible size (measure zero). Third, work will be done on existence, multiplicity and asymptotics for combustion theory models with complex chemistry including radicals. The models represent steady planar flame fronts in premixed combustion. This analysis goes beyond traditional simple reactions. Most models have to confront or hypothesize away what is known as the cold boundary difficulty to support traveling wave solutions. If intermediate species or radicals are introduced into a system of reactions, a scheme of the principal investigator leads to new existence proofs for flame layers. Work will proceed in expanding methods developed to cover other complex models, validate their asymptotic expansions and consider problems of multiplicity or uniqueness.

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences. It is for the purchase of special purpose equipment dedicated to the support of research in the mathematical sciences. In general, this equipment is required by several research projects, and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers, and with substantial cost-sharing from the institution submitting the proposal. This is an instance of university, industrial, and government cooperation in the support of basic research in the mathematical sciences. The equipment in this project will be used to support research on Coupled Oscillators and Neural Networks; Travelling Waves in Combustion Theory; On the Relation between Painleve Transcendents and Inverse Scattering; Nonlinear Boundary Value Problems in Ordinary Differential Equations; and On Systems of Differential Equations which Describe Phase Boundaries.

The topic of this conference is various aspects of similarity solutions of differential equations (solutions obtained by the classical Lie group method or generalizations), the occurrence of similarity solutions in applicable situations, the study of similarity solutions of particular equations, and the relation of similarity solutions to more general solutions of the relevant equations. Subjects to be discussed include travelling waves for reaction-diffusion equations, blow-up and quenching problems, equations soluble by inverse scattering and the Painleve transcendent equations, and boundary layer problems. This grant will help support a conference to be held at the University of Pittsburgh on April 26-28, 1991, whose goals are to provide a forum for the exchange of ideas and results on the leading edge of research in this topic.

9302737 Hastings This project concerns investigations into mathematical problems of nonlinear ordinary differential equations. There are six areas under investigation. They include asymptotics and painleve transcendents involving solutions obtained by inverse scattering, shooting methods applied to the study of chaos, specially in the analysis of Lorenz equations, and self-similar solutions of Barenblatt's fourth order model for the decay of turbulent bursts. Work will also be done expanding recent studies of stability in modeling solid state diffusion in semiconductors, asymptotics beyond all orders in fifth order equations and on the smoothing of Stokes discontinuities in the asymptotic expansions of analytic functions. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

Hastings 9703630 Semi-discrete models of excitable media consist of coupled systems of ordinary differential equations, with each unit describing the electrical kinetics in a single "cell". They are intermediate between continuous (pde) models and purely discrete models, or cellular automata. The investigator studies whether principles developed for cellular automata models apply to the more realistic semi-discrete case. He considers specific models developed from experimental data to give quantitative descriptions of neural behavior. A model of special interest is that of Morris and Lecar, describing voltage oscillations in barnacle muscle fiber. This model has different features from the well-known FitzHugh-Nagumo system, features that are found in a number of experimental settings and that, moreover, appear amenable to mathematical analysis. The investigator explores the role of various parameters in the system in the propagation of spatial patterns. Among the factors to be considered are excitation threshold, coupling strength, relative rates of fast and slow processes, spatial geometry of the medium, and the relative time spent in the excited and refractory states. The goal of this project is to develop an understanding of pattern formation in semi-discrete excitable media. The main intended application is to neurobiology, though excitable media are found in many biological and chemical systems. Isolated cells are called ``excitable'' if they exhibit a threshold phenomenon in response to a brief external stimulus, but cannot support continued oscillations on their own. A semi-discrete model describes collections of excitable neurons coupled by a linear or nonlinear diffusion mechanism. Examples include nerve, cardiac, and muscle tissue, supporting phenomena such as wave propagation down a myelinated axon, waves of electrical stimulation that sweep through heart muscles, and spreading cortical depression, a brain wave phenomenon.

Hastings 9801227 The investigator and his colleagues organize an interdisciplinary conference to bring together biologists and mathematicians to discuss wave phenomena from their varying perspectives. The conference aims to give both biologists and mathematicians insights into the types of models that can be used for wave phenomena and the parameter ranges where such behavior can be expected. To this end, the conference includes both general lectures and more technical talks where particular techniques are explored more fully. Of special interest in techniques are continuation methods in models of long range interaction, where integral equation models are involved. One-dimensional traveling waves have long been of interest to biologists, particularly in neurobiology where they describe the propopagation of electrical signals down a nerve axon or as a plane wave across a two-dimensional collection of electrically active cells. Related phenomena include spiral and other patterns, such as those thought to be responsible for some pathogenic behavior in cardiac tissue. Similar patterns in the brain are of current interest as well. Such behavior is not limited to neurobiology, and appears in a wide variety of chemical and biological systems, such as the Belousov-Zhabotinsky reaction, slime molds, and many others. On the other hand, mathematicians have studied basic questions about waves for a variety of models, including biological and chemical settings. One focus of mathematical work has been to prove the existence and stability of traveling waves. In this regard continuation methods have become particularly interesting in models of long range interaction, where integral equations are involved. The conference brings together biologists and mathematicians to discuss wave phenomena from their different perspectives. The meeting fosters interactions between the two areas that should lead to greater understanding of a variety of phenomena important in biology.