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Breathers and rogue waves for semilinear curl-curl wave equations

Plum, Michael 1; Reichel, Wolfgang 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

Abstract:

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial _t^2 U +\nabla \times \nabla \times U + q(x) U \pm V(x) \vert U \vert ^{p-1} U = 0$ for $(x,t)\in {\mathbb {R}}^3\times {\mathbb {R}}$ and arbitrary p>1 . Depending on the coefficients s, q, V we can prove the existence of three types of localized solutions: time-periodic solutions decaying to 0 at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to 0 both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution U(x, t) already generates a full continuum of phase-shifted solutions U(x,t+b(x)) where the continuous function $b:{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ belongs to a suitable admissible family.


Verlagsausgabe §
DOI: 10.5445/IR/1000158503
Veröffentlicht am 05.05.2023
Originalveröffentlichung
DOI: 10.1007/s41808-023-00215-x
Dimensions
Zitationen: 1
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Zeitschriftenaufsatz
Publikationsmonat/-jahr 12.2023
Sprache Englisch
Identifikator ISSN: 2296-9020, 2296-9039
KITopen-ID: 1000158503
Erschienen in Journal of Elliptic and Parabolic Equations
Verlag Springer
Band 9
Heft 2
Seiten 757–780
Vorab online veröffentlicht am 26.04.2023
Nachgewiesen in Dimensions
Scopus
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