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Travelling waves for Maxwell’s equations in nonlinear and nonsymmetric media

Mederski, Jarosław; Reichel, Wolfgang

Abstract:

We look for travelling wave fields $$ E(x,y,z,t) = U(x,y) \cos(kz+\omega t) + \tilde{U}(x,y) \sin(kz+\omega t), \quad (x,y,z)\in\mathbb{R}^3, t\in\mathbb{R} $$
satisfying Maxwell’s equations in a nonlinear medium which is not necessarily cylindrically symmetric. The nonlinearity of the medium enters Maxwell’s equations by postulating a nonlinear material law $D = \varepsilon E + \chi(x,y,\langle|E|^2\rangle)E$ between the electric field $E$, its time averaged intensity $\langle|E|^2\rangle)$ and the electric displacement field $D$. We derive a new semilinear elliptic problem for the profiles $U, \tilde{U}:\mathbb{R}^2 \to \mathbb{R}^3$ $$ Lu - V(x,y)u = f(x,y,u) \quad \text{with } u=\left(\begin{array}{cc} U \\ \tilde{U} \end{array}\right), \text{ for } (x,y) \in \mathbb{R}^2 ,$$
where $f (x, y, u) = \omega^2 \chi(x, y, |u|^2 )u$. Solving this equation we can obtain exact travelling wave solutions of the underlying nonlinear Maxwell equations. We are able to deal with super quadratic and subcritical focusing effects, e.g. in the Kerr-like materials with the nonlinear susceptibility of the form $\chi(x, y, \langle|E|^2\rangle E) = \chi^{(3)} (x, y) \langle|E|^2\rangle E$. ... mehr


Volltext §
DOI: 10.5445/IR/1000141622
Veröffentlicht am 03.01.2022
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 01.2022
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000141622
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 31 S.
Serie CRC 1173 Preprint ; 2022/1
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter Maxwell equations, Kerr nonlinearity, curl-curl problem, travelling wave, variational methods
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