Surface waves as Fourier integral operators with complex phase

Abstract

This thesis studies linear hyperbolic boundary value problems that admit surface waves as solutions. Surface waves are related to a specific type of weakly regular hyperbolic boundary value problems, where the precise meaning of weak is to be determined. With this thesis, we aim to provide a theoretical framework for rigorously analyzing such problems. To this end, we show that, under appropriate assumptions, a solution of a hyperbolic boundary value problem can be approximated by an oscillatory integral with complex-valued phase function. Then, we use the theory of Fourier integral operators with complex phase to study the properties of that particular solution. Following this approach, we are able to provide a refined description of the propagation of singularities as well as a preliminary result concerning the regularity of the solution in the context of Sobolev spaces H^s. Furthermore, we present some original results that complement the existing theory of Fourier integral operators with complex phase. In particular, we propose an alternative construction of the principal symbol of the operators, and use it to compute the principal symbol after composition under the assumption of clean intersection.