This project is concerned with (1) the stability of nonlinear waves and the interaction between nonlinear waves in a variety of physical systems, and (2) the application of mathematical tools from computer-aided geometric design (CAGD) to problems in dynamical systems. The first project addresses symmetry-breaking bifurcations in equations modeling a physical system that can "trap" pulses of light. It will also look at so-called collective-coordinate models of solitary wave interactions - both the mathematical validity of this approach (still a major unsolved area) and some of the mathematical structures that arise in these models. This includes the application of new and powerful topological methods to simple iterated maps derived by the PI as approximations to the collective-coordinate dynamics. The second project will use recently developed spline tools as the basis for new algorithms for computing invariant manifolds of iterated maps and vector fields, which should greatly improve the speed, accuracy, and simplicity of such computations in comparison with current methods.
This project is concerned with (1) the stability of nonlinear waves and the interaction between nonlinear waves in a variety of physical systems, and (2) the application of mathematical tools from computer-aided geometric design (CAGD) to problems in dynamical systems. The first project deals with nonlinear waves that arise in the modeling of systems from many areas of physics and engineering, with an emphasis here on models derived from optics. It will study how and under what conditions such waves may become unstable, for example how they might break apart. It will also study what happens when such waves collide, applying powerful mathematical (topological) techniques to simplified models in order to gain a deeper understanding, as well as studying whether such reduced models are valid. The second project will study new, and more efficient, ways to compute invariant manifolds, which are fundamental objects in understanding chaos. It will integrate research from the field of computer graphics into a new context important for mathematical physics. The goal is to use types of curves and surfaces developed for application to software for designing automobiles and aircraft, and for computer animation used in movies and television, for an unexpected scientific purpose.
