9401248 Williams This award supports mathematical research on problems arising in the theory and application of nonlinear partialdifferential equations. Particular emphasis is placed on the use of microlocal methods in the study of hyperbolic equations, developing methods for understanding diffraction of singularities by smooth obstacles in several types of nonlinear wave problems, resonant reflection and transmission of continuous multidimensional oscillations. Work will also be done on nonlinear optics for multidimensional shocks. A major goal of the project is to introduce the consideration of boundaries into rigorous nonlinear geometrical optics, which has until now dealt mostly with problems in free space. The importance of boundaries is indicated, for example, by the fact that multidimensional shock front and vortex sheet problems can be formulated as nonlinear hyperbolic free boundary problems. Also, boundaries can produce striking effects (such as resonant reflection) even in continuous solutions. Partial 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. ***
