Limiting Amplitude Principle Research Articles

On the limiting amplitude principle for the wave equation with variable coefficients

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems.

Limiting amplitude principle and resonances in plasmonic structures with corners: Numerical investigation

The limiting amplitude principle states that the response of a scatterer to a harmonic light excitation is asymptotically harmonic with the same pulsation. Depending on the geometry and nature of the scatterer, there might or might not be an established theoretical proof validating this principle. In this paper, we investigate a case where the theory is missing: we consider a two-dimensional dispersive Drude structure with corners. In the non lossy case, it is well known that looking for harmonic solutions leads to an ill-posed problem for a specific range of critical pulsations, characterized by the metal’s properties and the aperture of the corners. Ill-posedness is then due to highly oscillatory resonances at the corners called black-hole waves. However, a time-domain formulation with a harmonic excitation is always mathematically valid. Based on this observation, we conjecture that the limiting amplitude principle might not hold for all pulsations. Using a time-domain setting, we propose a systematic numerical approach that allows to give numerical evidences of the latter conjecture, and find clear signature of the critical pulsations. Furthermore, we connect our results to the underlying physical plasmonic resonances that occur in the lossy physical metallic case.

The limiting amplitude principle for the nonlinear Lamb system

The limiting amplitude principle for the nonlinear Lamb system

Analysis in temporal regime of dispersive invisible structures designed from transformation optics

A simple invisible structure made of two anisotropic homogeneous layers is analyzed theoretically in temporal regime. The frequency dispersion is introduced and analytic expression of the transient part of the field is derived for large times when the structure is illuminated by a causal excitation. This expression shows that the limiting amplitude principle applies with transient fields decaying as the power -3/4 of the time. The quality of the cloak is then reduced at short times and remains preserved at large times. The one-dimensional theoretical analysis is supplemented with full-wave numerical simulations in two-dimensional situations which confirm the effect of dispersion.

On uniqueness and stability of Sobolev’s solution in scattering by wedges

We prove that the solution of the scattering problem of pulses, constructed by Sobolev (Proc Seismol Inst Acad Sci URSS 41:1–15, 1934), coincides with the solution obtained by our method of complex characteristics. The coincidence holds for the DD- and NN-boundary conditions. The method of complex characteristics has been developed by Komech and Merzon in 2002–2007. Its main advantage is that it provides (a) the existence and uniqueness of the solutions in suitable functional classes and (b) the limiting amplitude principle. The uniqueness in the Sobolev approach was not considered. Our result means that Sobolev’s solution belongs to our functional classes and agrees with the limiting amplitude principle.

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave.As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.

On the Keller‐Blank solution to the scattering problem of pulses by wedges

We prove that the solution of the scattering problem of pulses constructed by Keller and Blank in 1951 coincides with the solution obtained by the method of complex characteristics. The method was developed by Komech and Merzon in 2006‐2007. Its main advantage is that it provides the existence and uniqueness of solutions in suitable functional classes, and the limiting amplitude principle. On the other hand, the uniqueness in the Keller‐Blank approach was not studied before. Our result means that the Keller‐Blank solution belongs to our functional classes. We prove the coincidence for DD and NN‐boundary conditions. Moreover, we obtain the solution for the DN‐case. Copyright © 2014 John Wiley & Sons, Ltd.

An Explicit Formula for the Nonstationary Diffracted Wave Scattered on a NN-Wedge

We consider two dimensional nonstationary scattering of plane waves by a NN-wedge. We prove the existence and uniqueness of a solution to the corresponding mixed problem and we give an explicit formula for the solution. Also the Limiting Amplitude Principle is proved and a rate of convergence to the limiting amplitude is obtained.

On the convergence of the amplitude of the diffracted nonstationary wave in scattering by wedges

We make more precise the Limiting Amplitude Principle in the two-dimensional scattering of an incident plane harmonic wave by a wedge. We find the long-time asymptotic regime of convergence of the amplitude of the cylindrical wave diffracted by the vertex of a wedge to the limiting amplitude of the solution to the corresponding stationary problem. The asymptotics turns out to be uniform on compacta and depends on the magnitude of the wedge and the profile of the incident wave. The cases of Dirichlet-Dirichlet and Dirichlet-Neumann boundary conditions are considered.

DN-Scattering of a plane wave by wedges

We continue our study of a nonstationary scattering by wedges. In this paper we consider nonstationary scattering of plane waves by a ‘hard–soft’ wedge. We prove the uniqueness and existence of a solution to the corresponding DN-Cauchy problem in appropriate functional spaces. We also give the explicit form of the solution and prove the Limiting Amplitude Principle. Copyright © 2011 John Wiley & Sons, Ltd.

The point source method for inverse scattering in the time domain

AbstractMany recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.

Limiting Amplitude principle in the scattering by wedges

AbstractWe consider a nonstationary scattering of plane waves by a wedge. We prove that the Sommerfeld‐type integral, constructed in (Math. Meth. Appl. Sci. 2005; 28:147–183; Proc. Int. Seminar ‘Day on Diffraction‐2003’, University of St. Petersburg, 2003; 151–162), is a classical smooth solution from a functional space, and prove the Limiting Amplitude principle. Copyright © 2006 John Wiley & Sons, Ltd.

Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media

Consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity. The propagation speed is discontinuous at the interface of two unbounded media, and the interface may be an arbitrary and smooth surface locally. A Sommerfeld radiation condition is introduced for the acoustic propagator, and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator. Furthermore, we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator.

On the limiting absorption and amplitude principles for transmission problems with unbounded interfaces

We consider the transmission problem for second-order differential equations with unbounded interfaces. The spaces B and B* are used to prove the limiting absorption principle for the steady-state problem and to establish the resolvent estimate at low frequencies for the steady-state operator. These are then applied in showing the validity of the limiting amplitude principle for such an operator.

The limiting amplitude principle for the acoustic wave operators in two unbounded media

The limiting amplitude principle for the acoustic wave operators in two unbounded media

Limiting amplitude principle for acoustic propagators in perturbed stratified fluids

By use of the commutator method, we study the resolvent estimate at low frequencies of acoustic operators − c( x) 2 Δ in perturbed stratified fluids and prove the principle of limiting amplitude for such operators.

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

AbstractWe consider a domain Ω in ℝn of the form Ω = ℝl × Ω′ with bounded Ω′ ⊂ ℝn−l. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂ u + [(− ∂ −… − ∂)m + (− ∂ −… − ∂)m]u = fe−iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.

Appliation of the limiting amplitude principle for field computation in a stratified medium

Appliation of the limiting amplitude principle for field computation in a stratified medium

Resolvent estimates at low frequencies and limiting amplitude principle for acoustic propagators

Resolvent estimates at low frequencies and limiting amplitude principle for acoustic propagators

The limiting amplitude principle for the Schroedinger equation in domains with unbounded boundaries

The limiting amplitude principle for the Schroedinger equation in domains with unbounded boundaries