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Breather solutions for a quasilinear (1+1)-dimensional wave equation

Kohler, Simon; Reichel, Wolfgang

Abstract:
We consider the $(1 + 1)$-dimensional quasilinear wave equation $g(x)w_{tt} − w_{xx} + h(x)(w^3_t)_t = 0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with $g\not\equiv 0$ and $h(x) = \gamma\delta_0(x)$ with $\gamma\in\mathbb{R}\backslash\{0\}$ and $\delta_0$ the delta distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k = \frac{d^2}{dx^2}-k^2\omega^2g$ on $L^2(\mathbb{R})$ for all $k\in 2\mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitely given step potentials and periodic step potentials $g$. ... mehr

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Volltext §
DOI: 10.5445/IR/1000132263
Veröffentlicht am 30.04.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 04.2021
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000132263
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 20 S.
Serie CRC 1173 Preprint ; 2021/17
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter breather, quasilinear wave equation, multiplicity, variational methods, nonlinear Maxwell equations
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