# 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

 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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