A note on separation conditions of resonance sets in the instability analysis for high-frequency oscillations in geometric optics

TL;DR

This paper relaxes the separation conditions of resonance sets in the instability analysis of high-frequency oscillations in hyperbolic systems, allowing for intersections and still deriving the same instability criteria.

Contribution

It extends previous instability criteria by allowing certain intersections among resonance sets, broadening the applicability of the analysis in geometric optics.

Findings

Relaxed the separation conditions of resonance sets in instability analysis.

Established that the same instability criterion applies despite intersections.

Applied results to coupled Klein-Gordon systems with specific nonlinear interactions.

Abstract

In this paper, we study the instability of highly-oscillating solutions to semi-linear hyperbolic systems. A instability criterion was given in \cite{Lu} under rather strong separation conditions of resonance sets: coupled resonance sets are pairwise disjoint. Here we show that such separation conditions in \cite{Lu} can be relaxed: one of the coupled non-transparent resonance sets is allowed to intersect with at most two others. We obtain the same instability criterion as in \cite{Lu}. Finally, we give some applications to coupled Klein-Gordon systems with equal masses and nonlinear terms specified particularly, where on the intersections of resonance sets, the related interaction coefficients are non-transparent.