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Breather solutions for a quasi‐linear (1+1)‐dimensional wave equation

Kohler, Simon 1; Reichel, Wolfgang 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)


We consider the (1 + 1)-dimensional quasi-linear wave equation $𝑔(𝑥)𝑤_{𝑡𝑡} − 𝑤_{𝑥𝑥} + ℎ(𝑥)(𝑤^{3}_{𝑡} )_{𝑡} = 0$ on ℝ×ℝ that arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions.
Here $𝑔 ∈ 𝐿^{∞}(ℝ)$ is even with 𝑔 ≢ 0 and $ℎ(𝑥) = 𝛾 𝛿_{0}(𝑥)$ with 𝛾 ∈ ℝ∖{0} and $𝛿_{0}$ the delta-distribution supported in 0. We assume that 0 lies in a spectral gap of the operators $𝐿_{𝑘} = − \frac {d^{2}}{d𝑥^{2}} − 𝑘^{2}𝜔^{2}𝑔$ on $𝐿^{2}(ℝ)$ for all 𝑘 ∈ 2ℤ+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 explicitly given step potentials and periodic step potentials 𝑔. In these examples, we even find infinitely many distinct breathers.

Verlagsausgabe §
DOI: 10.5445/IR/1000140438
Veröffentlicht am 29.11.2021
DOI: 10.1111/sapm.12455
Zitationen: 1
Web of Science
Zitationen: 1
Zitationen: 2
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2021
Sprache Englisch
Identifikator ISSN: 0022-2526, 1467-9590
KITopen-ID: 1000140438
Erschienen in Studies in applied mathematics
Verlag John Wiley and Sons
Band 148
Heft 2
Seiten 689-714
Vorab online veröffentlicht am 11.10.2021
Nachgewiesen in Web of Science
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