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Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

  • Thomas Bartsch
  • Tomáš Dohnal
  • Michael Plum
  • Wolfgang ReichelEmail author
Article

Abstract

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.

Keywords

Curl-curl problem Maxwell’s equation Ground state Variational methods Symmetric subspace Concentration compactness 

Mathematics Subject Classification

Primary 35Q60 58E15 Secondary 47J30 78A25 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of GiessenGiessenGermany
  2. 2.Department of MathematicsTechnical University DortmundDortmundGermany
  3. 3.Institute for AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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