Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Mandel, Rainer; Moitier, Zoïs; Verfürth, Barbara

doi:10.5802/crmath.322

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…Since then, such problems have been studied in several contexts. For instance, we can refer the reader interested in the analysis of indefinite problems to [6] for scalar transmission problems, [18] for Helmholtz type problems, [7] for Maxwell's system, [24] for a non linear signchanging problem, [16] for the study of scattering resonances and [17,8] for numerical aspects. In the context of the homogenization for perfect transmission problems with sign-changing coefficients, the T−coercivity has been used in [12,13,14] for scalar diffusion problems and in [11] for Maxwell's system.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump

Bunoiu,

Ramdani,

Timofte

2023

Communications in Mathematical Sciences

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We study the homogenization of a scalar problem posed in a composite medium made up of two materials, a positive and a negative one. An important feature is the presence of a flux jump across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and to the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction (nonstandard in this context) and T−coercivity technics are used in order to cope with these difficulties.

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Section: Introductionmentioning

confidence: 99%

Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump

Bunoiu,

Ramdani,

Timofte

2023

Communications in Mathematical Sciences

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Scattering resonances in unbounded transmission problems with sign-changing coefficient

Carvalho

Moitier

2023

IMA Journal of Applied Mathematics

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It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the “quasimodes to resonances” argument from the black box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial’s properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided.

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Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Moitier

2022

Preprint

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In this talk, we are interested in the combine effect of metamaterial and Kerr non-linearity. More precisely, we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains of the form -div (σ(x) ∇ u) - λ u = u^3. Using weak 𝚃-coercivity theory, we can establish the existence of an orthonormal basis of eigenfunctions of the linear part -div (σ(x) ∇ u). Then, all eigenvalues are proved to be bifurcation points, and we investigate the bifurcating branches both theoretically and numerically. As a fundamental example, we look at some one-dimensional model, we obtain the existence of infinitely many bifurcating branches that are mutually disjoint, unbounded, and consist of solutions with a fixed nodal pattern.

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Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Cited by 3 publications

References 21 publications

Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump

Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump

Scattering resonances in unbounded transmission problems with sign-changing coefficient

Nonlinear Helmholtz equations with sign-changing diffusion coefficient

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