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

Two-layered discrete waveguide is considered. The governing equations are matrix Klein-Gordon equations of dimension 2. The formal solution of this system in terms of double integral can be obtained using Fourier transform on space and time variables. Then, the double integral can be reduced to single integral with the help of residue integration on frequency or wavenumber variable. However, such an integral can be di cult to estimate since it involves branching and oscillating functions. Instead, we study it asymptotically and classify each asymptotic by the amount of singular points in the domain on influence. Then we present a zone diagram. Each zone of the diagram is characterized by its own asymptotic. On the boundary between zones we establish a parent / child relation. We generalize the common diffraction concept of the far-field zone and the near-field zone. Based on the zone diagram, it is shown that the field can be approximately represented as sum of two mode pulses and an exchange pulse. New special function to describe the latter is introduced. The work is supported by the RFBR grant 19-29-06048.