Rigorose Asymptotik von Surface-Plasmon-Polariton-Wellenpaketen in nichtlinearen Medien
Zusammenfassung der Projektergebnisse
Surface plasmon polaritons (SPPs) at an interface of a metal and a dielectric are modelled by Maxwell equations with terms non-local in time (so called material dispersion) and discontinuous in space due to the interface. We studied nonlinear materials, which have the potential of supporting coherent localized wavepackets. The aim of the proposal was to extend the mathematical understanding of this system and to study wavepackets of SPPs in a slowly varying envelope approximation. We firstly derived and justified the asymptotics of time dependent wavepackets in the transverse magnetic polarization at the interface of two non-dispersive media in two spatial dimensions. Only local well-posedness of this quasi-linear Maxwell system had been known and in collaboration with Roland Schnaubelt (Karlsruhe) we extended it to asymptotically large time intervals for small data. The well-posedness of the quasilinear three dimensional Maxwell system with material dispersion was studied in collaboration with Marcus Waurick (Freiberg). Using Bochner spaces exponentially weighted in time, we proved global well-posedness of the linear problem under accretivity conditions on the dielectric constant. This was then extended to several nonlinearities which allow Lipschitz continuity in the Bochner space. In addition, we proved exponential stability results on bounded domains. The exponential stability is crucial for the justification of the asymptotic approximation as the error must stay small on asymptotically large time intervals. The limitation to bounded domains currently prevents us from applying the theory to SPP wavepackets in dispersive media because such wavepackets exist in unbounded domains. For the linear time harmonic problem with homogeneous media on each side of the interface we have determined the full spectrum of the operator in one and two spatial dimensions - with the frequency ω playing the role of the spectral parameter. As the dielectric constant depends nonlinearly on ω, one has to analyse an operator pencil. In collaboration with M. Brown (Cardiff), M. Plum (Karlsruhe), and I. Wood (Canterbury) we analysed the spectrum for the interface problem with a homogeneous material in each half space. In the nonlinear time harmonic problem, in collaboration with G. Romani (Halle), we proved the existence of localized solutions (bound states) bifurcating from simple isolated eigenvalues of the linear problem. This is again a bifurcation in an operator pencil setting as ω plays the role of a bifurcation parameter. We showed that in the transverse electric setting there are geometries with three material layers which support simple isolated eigenvalues. We provided an asymptotic expansion of the bifurcating bound state. Under the so called PT-symmetry of the geometry with a real eigenvalue the bifurcation parameter has been shown to remain real at least locally. This produces a conservative (lossless) time harmonic SPP bound state. The bifurcation problem for the transverse magnetic setting is an ongoing project.
Projektbezogene Publikationen (Auswahl)
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Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons. Nonlinear Differential Equations and Applications NoDEA, 28(1).
Dohnal, Tomáš & Romani, Giulio
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Justification of the asymptotic coupled mode approximation of out-of-plane gap solitons in Maxwell equations. Nonlinearity, 34(8), 5261-5318.
Dohnal, Tomáš & Romani, Giulio
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A quasilinear transmission problem with application to Maxwell equations with a divergence-free D-field. Journal of Mathematical Analysis and Applications, 511(1), 126067.
Dohnal, Tomáš; Romani, Giulio & Tietz, Daniel P.
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Rigorous Envelope Approximation for Interface Wave Packets in Maxwell’s Equations with Two Dimensional Localization. SIAM Journal on Mathematical Analysis, 55(6), 6898-6939.
Dohnal, Tomáš; Schnaubelt, Roland & Tietz, Daniel P.
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Spectrum of the Maxwell Equations for a Flat Interface Between Homogeneous Dispersive Media. Communications in Mathematical Physics, 406(1).
Brown, Malcolm; Dohnal, Tomáš; Plum, Michael & Wood, Ian
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Well-posedness and exponential stability of nonlinear Maxwell equations for dispersive materials with interface. Journal of Differential Equations, 383, 24-77.
Dohnal, Tomáš; Ionescu-Tira, Mathias & Waurick, Marcus
