The interaction of nonlinear hyperbolic waves

Nonlinearities in wave equations lead to focusing and defocusing of solutions. Focusing causes sharply defined wave fronts. The interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as Mach triple point formation, shock wave diffraction patterns and the study of Riemann problems in one and higher dimensions. Recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures. This mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations. It is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena. Within the subject matter and point of view adopted here, we have attempted to present a broad and, we hope, a representative account of recent progress.