Solving the Scattering Problem for Open Wave-Guide Networks, II Outgoing Estimates

TL;DR

This paper analyzes the asymptotic behavior of solutions to a wave-guide scattering problem, establishing outgoing radiation conditions and ensuring solution uniqueness through integral equation methods.

Contribution

It demonstrates that solutions to the integral equations have asymptotic expansions, leading to outgoing radiation conditions and uniqueness results for the wave-guide scattering problem.

Findings

Solutions admit asymptotic expansions under certain data conditions.

Outgoing radiation conditions are satisfied by the solutions.

Uniqueness of solutions is established through these conditions.

Abstract

The paper continues the analysis, started in [1] (Part I,arXiv:2302.04353), of the model open wave-guide problem defined by 2 semi-infinite, rectangular wave-guides meeting along a common perpendicular line. In Part I we reduce the solution of the physical problem to a transmission problem rephrased as a system of integral equations on the common perpendicular line. In this part we show that solutions of the integral equations introduced in Part I have asymptotic expansions, if the data allows it. Using these expansions we show that the solutions to the PDE found in each half space satisfy appropriate outgoing radiation conditions. In Part III we show that these conditions imply uniqueness of the solution to the PDE as well as uniqueness for our system of integral equations.