Complex geometric optics for symmetric hyperbolic systems I: linear theory

TL;DR

This paper develops a rigorous linear theory for complex geometric optics applied to symmetric hyperbolic systems with oscillatory initial data, establishing foundational results for future nonlinear analysis in wave propagation models.

Contribution

It introduces the first rigorous linear framework for complex geometric optics in symmetric hyperbolic systems with complex phases, paving the way for nonlinear extensions.

Findings

Asymptotic solutions for hyperbolic systems with complex phases

Foundation for nonlinear complex geometric optics

Rigorous linear theory established

Abstract

We obtain an asymptotic solution for $\ep \to 0$ of the Cauchy problem for linear first-order symmetric hyperbolic systems with oscillatory initial values written in the eikonal form of geometric optics with frequency $1/\ep$, but with complex phases. For the most common linear wave propagation models, this kind on Cauchy problems are well-known in the applied literature and their asymptotic theory, referred to as complex geometric optics, is attracting interest for applications. In this work, which is the first of a series of papers dedicated to complex geometric optics for nonlinear symmetric hyperbolic systems, we develop a rigorous linear theory and set the basis for the subsequent nonlinear analysis.