Stability of Discrete Breathers in Magnetic Metamaterials
We consider the discrete Klein–Gordon equation for magnetic metamaterials derived by Lazarides, Eleftheriou, and Tsironis Phys Rev Lett 97:157406, 2006). We obtain a general criterion for spectral stability of multi-site breathers for a small coupling con
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Abstract We consider the discrete Klein–Gordon equation for magnetic metamaterials derived by Lazarides, Eleftheriou, and Tsironis Phys Rev Lett 97:157406, 2006). We obtain a general criterion for spectral stability of multi-site breathers for a small coupling constant. We show how this criterion differs from the one derived in the standard discrete Klein–Gordon equation (Koukouloyannis and Kevrekidis, Nonlinearity 22:2269–2285, 2009; Pelinovsky and Sakovich, Nonlinearity 25:3423–3451, 2012).
1 Introduction We address space-localized and time-periodic breathers in the discrete Klein– Gordon equation describing magnetic metamaterials which consist of periodic arrays of split-ring resonators [4, 7]: qRn C V 0 .qn / D .qRnC1 C qR n1 /;
n 2 Z;
(1)
where t 2 R is the evolution time, qn .t/ 2 R is the normalized charge stored in the capacitor of the n-th split-ring resonator, V W R ! R is a smooth on-site potential for the voltage across the slit of the n-th resonator, and 2 R is the coupling constant from the mutual inductance. In particular, the voltage u D f .q/ D V 0 .q/ D. Pelinovsky () Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada e-mail: [email protected] V. Rothos Faculty of Engineering, School of Mathematics, Physics and Computer Sciences, Aristotle University, GR54124 Thessaloniki, Greece 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__18, © Springer International Publishing Switzerland 2014
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is found by inverting the charge-voltage dependence near small charge: q D u C ˛u3
)
u D f .q/ D q ˛q 3 C O.q 5 /
as q ! 0;
(2)
where ˛ is the parameter for the self-focusing (˛ > 0) or self-defocusing (˛ < 0) nonlinearity. These parameter values correspond to the soft and hard potentials V respectively, for sufficiently small values of q. Note that V is an even function of q. Discrete breathers in both one-dimensional and two-dimensional lattices were approximated numerically in the limit of small coupling constant [4,7]. Excitations of discrete breathers near the edge of a one-dimensional lattice created by a truncated array of nonlinear split-ring resonators were considered numerically in [8]. It is the purpose of this paper to consider spectral stability of multi-site discrete breathers in the limit of small coupling constant . This limit is referred usually as the anti-continuum limit and it has been considered before in the context of spectral stability of discrete breathers in the standard discrete Klein–Gordon equation [1,3,6,10] and in the discrete nonlinear Schrödinger equation [11,13]. Recent works [12, 14, 16] were devoted to the derivation of the most general stability criterion for multi-site breathers in Klein–Gordon lattices. Our main result shows that the stability criterion for multi-site breathers in the discrete Klein–Gordon equation (1) differs from the one derived
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