Travelling Waves in Nonlinear Magnetic Metamaterials
In this article, a model of one-dimensional metamaterial formed by a discrete array of nonlinear resonators is considered. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented. The existence and the
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stract In this article, a model of one-dimensional metamaterial formed by a discrete array of nonlinear resonators is considered. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented. The existence and the stability of asymptotic waves are also computed and discussed numerically.
J. Diblík Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic e-mail: [email protected] M. Feˇckan () Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia e-mail: [email protected] M. Pospíšil Centre for Research and Utilization of Renewable Energy, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic e-mail: [email protected] V.M. Rothos Department of Mathematics, Physics and Computational Sciences, Mathematics Division, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki GR54124, Greece e-mail: [email protected] H. Susanto School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__17, © Springer International Publishing Switzerland 2014
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1 Introduction Metamaterials are artificial materials that are engineered to have properties that may not be found in nature. The modification is achieved by composing inhomogeneities, i.e. structural rather than chemical, to create desirable effective behaviour. Primarily the study is on engineering the refractive index of a material. Since the proposal [21], a new paradigm in electromagnetism has emerged due to these new types of artificial composites, including cloaking devices [22] (see also [18] for a review of recent results for electromagnetic manipulation enabled by metamaterials). The canonical constituent components, or meta-atoms from which metamaterials are fashioned are the split ring resonator, which consists of an inductive metallic ring with a gap to provide capacitance, or its U -shaped modifications. Previous studies of metamaterials were focused on the linear properties of the medium during wave propagation, in which case magnetic permeability and material permittivity are non-dependent on the intensity of the electromagnetic field. When nonlinearity is present in metamaterials due to either employing a nonlinear host medium [20] or by engineering the elements of a metamaterial with a nonlinear component [13], nontrivial properties and behaviours can be present, i.e. dynamic tunable systems and active artificial media, such as materials that are compressible by magnetic field [15] and
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