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A breather construction for a semilinear curl-curlwave equation with radially symmetric coefficients

Plum, Michael; Reichel, Wolfgang


Abstract. We consider the semilinear curl-curl wave equation s(x)∂2 U + ∇ × ∇ × U + q(x)U ± V (x)|U |p−1 U = 0 for (x, t) ∈ R3 × R. For any p > 1 we prove the existence of time- periodic spatially localized real-valued solutions (breathers) both for the + and the − case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U (x, t+a(x)), where U is a particular breather and a : R3 → R an arbitrary radially symmetric C 2 -function.

Volltext §
DOI: 10.5445/IR/1000060946
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2016
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000060946
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 9 S.
Serie CRC 1173 ; 2016/29
Schlagwörter semilinear wave-equation, breather, phase plane method
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