Reflection of conormal pulse solutions to large variable-coefficient semilinear hyperbolic systems

TL;DR

This paper rigorously justifies nonlinear geometric optics expansions for reflecting pulses in variable-coefficient semilinear hyperbolic systems, including complex boundary interactions and multiple pulse interactions in higher dimensions.

Contribution

It extends previous work by removing the coherence assumption and applies to systems with multiple interacting pulses and nonlinear phases, covering boundary reflection problems.

Findings

Validates geometric optics expansions for pulse reflections in hyperbolic systems.

Handles multiple interacting pulses with nonlinear phases.

Applicable to boundary and free space problems in higher dimensions.

Abstract

We provide a rigorous justication of nonlinear geometric optics expansions for reflecting \emph{pulses} in space dimensions $n>1$. The pulses arise as solutions to variable coefficient semilinear first-order hyperbolic systems. The justification applies to $N\times N$ systems with $N$ interacting pulses which depend on phases that may be nonlinear. The \emph{coherence} assumption made in a number of earlier works is dropped. We consider problems in which incoming pulses are generated from pulse boundary data as well as problems in which a single outgoing pulse reflects off a possibly curved boundary to produce a number of incoming pulses. Although we focus here on boundary problems, it is clear that similar results hold by similar methods for the Cauchy problem for $N\times N$ systems in free space.