Summary Article
Mathematics -- The Science of Patterns and Algorithms
Nonlinearity
Understanding nonlinearity is a central challenge for modern science. This is a huge task, since there are many sources and many sorts of nonlinearity. Sometimes its origin is the behavior being modeled: physical examples include combustion, phase transformation, and turbulence; biological examples include protein folding and excitable tissues, such as heart muscle and the nervous system. Nonlinearity can also come from other, more structural sources such as feedback or geometry: examples include the optimization of financial decisions; the pinch-off of fluid droplets; and the motion of surfaces under curvature-driven flows.
We have accumulated great insight into nonlinear phenomena, and a rich collection of viewpoints and methods, such as asymptotic analysis, bifurcation, chaos, shock waves, and viscosity solutions, to name but a few. Our powerful and ever-expanding collection of tools provides many opportunities for advancement. For example, new methods from the calculus of variations are giving fresh insight into the design of composites with optimal microstructures, the consequences of polycrystalline structure in shape-memory materials, and the arrangement of domain walls in magnetic materials [KO]. Another example: nonlinear evolution equations adapted from geometry and fluid dynamics are being used for the creation, manipulation, smoothing, and segmentation of visual images [CH], [MU], [EV]. A third example: dynamical systems methods are providing insight into the sources and consequences of spatio-temporal patterns, including fibrillation of heart tissue and rhythmic electrical activity in the nervous system [KL].
And yet our mastery of nonlinearity has barely begun. Many fundamental questions remain open, for example: Are solutions of the Navier-Stokes for fluid mechanics equations unique in three space dimensions? What about (viscosity) solutions of systems of hyperbolic conservation laws? These questions are mathematical challenges of course, but their importance goes much deeper. They address the adequacy of our standard mathematical models for fluid mechanics and gas dynamics. If solutions were not unique--which would be a big surprise--then some effect we usually ignore would actually be crucial for determining the physically correct solution. If, as seems more likely, solutions are unique, then the methods developed to prove this assertion will also give insight concerning qualitative features of flows and the accuracy and stability of numerical solution schemes. Recent breakthroughs on uniqueness for conservation laws make this topic particularly timely [EV].