Mathematical Sciences: Geometric Approaches to the Stability of Travelling Waves & Other Problems

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Travelling waves are the solutions of partial differential equations that transport information in a fixed direction. A physically observable wave must be stable to perturbations in its profile. The work under this grant will be aimed at the development of techniques for discriminating betwen stable and unstable waves. In diffusive problems, there is envisioned a theory in which the structure of the wave determines any instabilities. The analysis of such systems necessarily involves a reduction to a simpler system, such as through a singular perturbation. A new geometric approach to singular perturbations is being developed with Kopell which is designed to solve stability problems in such equations as those of Hodgkin-Huxley for nerve impulse propagation. Other stability problems to be attacked involve the subtle effects ocurring in fluid and optical problems. Exponential decay to waves does not happen in these cases and new techniques need to be developed. The theory of transversality in dynamical systems will also be developed and aplied to problems in spin-orbit resonance and uniqueness in elliptic problems. The complexity of physical systems that can be analyzed is confined by the mathematical techniques available. Systems with some simple character, such as those which are linear or an equation with a single unknown function, yield to well-developed methods of analysis that are now part of the engineer's toolbox. However, more complicated systems remain a challenge to the best applied mathematicians and many important phenomena are missed by the simplification to systems that can be easily analyzed. Future advances in science may well be driven by the mathematical development of techniques for the analysis of such complicated systems. One aspect of such a system that demands a mathematical solution is the question of the stability of basic structures, under the perturbing influences suplied by the outside world. The work in this grant will be largely devoted to the development of techniques for assessing the stability of such structures. It is hoped that at some point in the future these techniques will be in the engineer's toolbox.