This IMA Volume in Mathematics and its Applications MICROLOCAL ANALYSIS AND NONLINEAR WAVES is based on the proceedings of a workshop which was an integral part of the 1988- 1989 IMA program on "Nonlinear Waves". We thank Michael Beals, Richard Melrose and Jeffrey Rauch for organizing the meeting and editing this proceedings volume. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. A vner Friedman Willard Miller, Jr. PREFACE Microlocal analysis is natural and very successful in the study of the propagation of linear hyperbolic waves. For example consider the initial value problem Pu = f E e'(RHd), supp f C {t ;::: O} u = 0 for t < o. If P( t, x, Dt,x) is a strictly hyperbolic operator or system then the singular support of f gives an upper bound for the singular support of u (Courant-Lax, Lax, Ludwig), namely singsupp u C the union of forward rays passing through the singular support of f.

Inhalt
On the interaction of conormal waves for semilinear wave equations.- Regularity of nonlinear waves associated with a cusp.- Evolution of a punctual singularity in an Eulerian flow.- Water waves, Hamiltonian systems and Cauchy integrals.- Infinite gain of regularity for dispersive evolution equations.- On the fully non-linear Cauchy problem with small data. II.- Interacting weakly nonlinear hyperbolic and dispersive waves.- Nonlinear resonance can create dense oscillations.- Lower bounds of the life-span of small classical solutions for nonlinear wave equations.- Propagation of stronger singularities of solutions to semilinear wave equations.- Conormality, cusps and non-linear interaction.- Quasimodes for the Laplace operator and glancing hypersurfaces.- A decay estimate for the three-dimensional inhomogeneous Klein-Gordon equation and global existence for nonlinear equations.- Interaction of singularities and propagation into shadow regions in semilinear boundary problems.