A note on the validity of the Schrödinger approximation for the Helmholtz equation

• Autorzy: Klumpp Maximilian , Schneider Guido

Identyfikatory

DOI

10.1515/jaa-2021-2058

• Języki publikacji: EN

Abstrakty

EN

Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δu + ω²u = 0 for (x, y, z) ∈ ℝ³. For the evolution of such waves along the z-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the z-axis.

Słowa kluczowe

EN

error estimates; optics

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2022
• Tom: Vol. 28, nr 1
• Strony: 67--72
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Klumpp Maximilian

• Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

autor

Schneider Guido

• Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Bibliografia

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• [7] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Grad. Stud. Math. 133, American Mathematical Society, Providence, 2012.
• [8] J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, SPIE Press, Bellingham, 2010.
• [9] G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances, J. Differential Equations 216 (2005), no. 2, 354-386.
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Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).