Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle

TL;DR

The paper develops a robust limiting absorption principle (LAP) for the one-dimensional Helmholtz problem with quasiperiodic coefficients by replacing Dirichlet-to-Neumann (DtN) transpare nt conditions with Robin-to-Robin (RtR) boundary data. A two-dimensional lifting and a fibered, periodic structure lead to a Riccati system that governs the propagation and scattering operators, which are analyzed spectrally under Diophantine irrationality conditions on the quasiperiodicity parameter. Under technical assumptions (notably non-Liouville irrationality and nonvanishing energy flux for the fundamental mode), the authors prove convergence of absorbing solutions to a physically meaningful limit, characterize the limit operators, and provide a numerically implementable procedure to compute the physical solution. They also connect the limiting behavior to spectral properties of the quasiperiodic operator, evanescent vs propagative regimes, and dispersion laws, with explicit links to flux densities and group velocity. The resulting framework yields both theoretical LAP results and a constructive algorithm suitable for simulations of wave propagation in quasi-periodic 1D media.

Abstract

We study the one-dimensional Helmholtz equation with (possibly perturbed) quasiperiodic coefficients. Quasiperiodic functions are the restriction of higher dimensional periodic functions along a certain (irrational) direction. In classical settings, for real-valued frequencies, this equation is generally not well-posed: existence of solutions in L 2 is not guaranteed and uniqueness in L $\infty$ may fail. This is a well-known difficulty of Helmholtz equations, but it has never been addressed in the quasiperiodic case. We tackle this issue by using the limiting absorption principle, which consists in adding some imaginary part (also called absorption) to the frequency in order to make the equation well-posed in L 2 , and then defining the physically relevant solution by making the absorption tend to zero. In previous work, we introduced a definition of the solution of the equation with absorption based on Dirichlet-to-Neumann (DtN) boundary conditions. This approach offers two key advantages: it facilitates the limiting process and has a direct numerical counterpart. In this work, we first explain why the DtN boundary conditions have to be replaced by Robin-to-Robin boundary conditions to make the absorption go to zero. We then prove, under technical assumptions on the frequency, that the limiting absorption principle holds and we propose a numerical method to compute the physical solution.

Figures (13)

• Figure 1: Function $F: ({y}_1, {y}_2) \mapsto \cos 2\pi {y}_1 + \cos 2\pi {y}_2$ in its periodicity cell $(0,1)^2$ (left), and whose trace along ${\boldsymbol{\theta}} = (1, \sqrt{2})$ leads to a quasiperiodic function (right).
• Figure 2: Representation of the set $\{(\{\theta_1\, {x}\}, \{\theta_2\, x\}),\ x \in (0, \tau)\}$ in $(0, 1)^2$ for different values of $\tau$, when $\theta_1/\theta_2 \in \mathbb{Q}$ (first row), and when $\theta_1/\theta_2 \in \mathbb{R} \setminus \mathbb{Q}$ (second row for ${\boldsymbol{\theta}} = (\sqrt{2}, 1)$ and third row for ${\boldsymbol{\theta}} = (\pi, 1)$).
• Figure 3: Notation introduced in Section \ref{['sub:Robin2D_eps']}
• Figure 4: The locally perturbed quasiperiodic coefficients $\mu$ and $\rho$, and the source term $f$.
• Figure 5: Position of the evanescent frequencies and of the propagative frequencies inside the spectrum of the quasiperiodic differential operator

Theorems & Definitions (116)

• Remark 1.1
• Theorem 1.2: Kronecker's theorem
• Remark 1.3
• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• Remark 2.5
• Corollary 2.6
• Remark 2.7