Asymptotical study of two-layered discrete waveguide with a weak coupling

TL;DR

This paper investigates the asymptotic behavior of solutions to a matrix Klein-Gordon equation modeling a two-layered waveguide with weak coupling, introducing a zone diagram technique to classify asymptotic regimes.

Contribution

It develops a novel zone diagram method to analyze asymptotics of a matrix Klein-Gordon equation in a layered waveguide, extending the understanding of wave propagation zones.

Findings

Zone diagram generalizes far-field and near-field zones.

Asymptotic formulas are derived for the waveguide solutions.

The method aids in estimating complex oscillatory integrals.

Abstract

A thin two-layered waveguide is considered. The governing equations for this waveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution of this system in the form of a double integral can be obtained by using Fourier transformation. Then, the double integral can be reduced to a single integral with the help of residue integration with respect to the time frequency. However, such an integral can be difficult to estimate since it involves branching and oscillating functions. This integral is studied asymptotically. A zone diagram technique is proposed to represent the set of possible asymptotic formulae. The zone diagram generalizes the concept of far-field and near-field zones.