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

Charles L. Epstein

TL;DR

This paper derives uniform, high-order asymptotic expansions for the open wave-guide network scattering problem of two semi-infinite rectangular channels meeting along a perpendicular line. By refining Part I’s integral-equation formulation on the common line and employing stationary-phase and contour-deformation techniques, the authors obtain precise decay and far-field expansions for the densities $(\sigma, au)$ and for the scattered fields $u^{l,r}$, including a decomposition into radiation and wave-guide-mode components. They establish uniform radiation conditions and show that the integral-equation system has trivial null-space under suitable function-space assumptions, linking the PDE solution to the limiting absorption framework. These results lay the groundwork for Part III’s proofs of uniqueness and for confirming the physical–mathematical consistency of the scattering problem in open wave-guide networks.

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.

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

TL;DR

This paper derives uniform, high-order asymptotic expansions for the open wave-guide network scattering problem of two semi-infinite rectangular channels meeting along a perpendicular line. By refining Part I’s integral-equation formulation on the common line and employing stationary-phase and contour-deformation techniques, the authors obtain precise decay and far-field expansions for the densities and for the scattered fields , including a decomposition into radiation and wave-guide-mode components. They establish uniform radiation conditions and show that the integral-equation system has trivial null-space under suitable function-space assumptions, linking the PDE solution to the limiting absorption framework. These results lay the groundwork for Part III’s proofs of uniqueness and for confirming the physical–mathematical consistency of the scattering problem in open wave-guide networks.

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.
Paper Structure (14 sections, 15 theorems, 305 equations, 7 figures)

This paper contains 14 sections, 15 theorems, 305 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Asymptotic Expansions
  3. Some Notation
  4. Outgoing Estimates for $\sigma$ and $\tau$
  5. The basic estimate
  6. The asymptotic expansion
  7. Asymptotics for The Free Space Part of $u^{l,r}$
  8. Estimates with $\eta_1\neq 0$
  9. Uniform Estimates as $\eta_1\to 0$
  10. Estimates for $\mathcal{W}^{l,r}\tau-\mathcal{W}^{l,r\,'}\sigma$
  11. Large $|y_2|$ contributions
  12. Small $|y_2|$ contributions
  13. Estimates with $x_1\to\infty,$ and $|x_2|$ bounded
  14. Concluding Remarks

Key Result

Theorem 1

If $\sigma,$ resp. $\tau,$ belongs to $\mathcal{C}_{\alpha}(\mathbb R),$ resp. $\mathcal{C}_{\alpha+\frac{1}{2}}(\mathbb R),$ for $0<\alpha<\frac{1}{2},$ has an asymptotic expansion like that given in eqn5.205, then the layer potential $v(r\eta),$ resp. $u(r\eta),$ has an asymptotic expansion, as $r

Figures (7)

  • Figure 1: Two dielectric channels meeting along a straight interface. The $x_3$-axis is orthogonal to the plane of the image.
  • Figure 2: Plots of $\Lambda_{\mu}$ and the deformations needed to analyze the integrals in \ref{['eqn259.101']} and \ref{['eqn267.200']}. The green curves are the right, resp. left boundary of $Q_+.$
  • Figure 3: Contour deformations for the integral in \ref{['eqn300.203']}. The deformation $\Lambda^{\theta}_{\mu+}$ includes the blue curve; the stationary point is the black dot.
  • Figure 4: Contour deformations for the integral in \ref{['eqn306.203']}. The deformation $\Lambda^{\theta}_{\mu-}$ includes the blue curve.
  • Figure 5: The contour $\Gamma^+_{\nu}$ shown in blue. The roots of Wronskian $\{\pm\xi_n\}$ are shown as red asterisks.
  • ...and 2 more figures

Theorems & Definitions (37)

  • Theorem : Theorem \ref{['thm3.75']}
  • Lemma : Lemma \ref{['lem5.202']}
  • Remark 1
  • Lemma 1
  • Remark 2
  • proof
  • Remark 3
  • Theorem : Theorem 3.2 (b), Chapter 5 of CoddingtonLevinson
  • Proposition 1
  • Remark 4
  • ...and 27 more