Hysteresis of Switching Waves and Dissipative Solitons in Nonlinear Magnetic Metamaterials

743 Artificial composite structures containing electric conducting elements or metamaterials have recently attracted much attention in view of their unique propp erties of negative magnetic susceptibility and backk ward wave propagation. In contrast to crystals, metamaterials allow control of the macroscopic charr acteristics by choosing the types and geometry of their structural elements [1–3]. Resonant magnetic metamaterials are most simply obtained by creating a periodic lattice of resonant electric circuits that are much smaller than the wavelength of propagating electromagnetic waves. The artificial character of metamaterials makes it possible to control their propp erties through either the dynamic rearrangement of their structure or the inclusion of additional nonlinear elements and control their properties by an external field [4]. It has recently been shown that magnetic metamaa terials consisting of chains of electric oscillatory cirr cuits—split ring resonators—can exhibit discreteness effects owing to their weak coupling [5–8]. In particuu lar, it was demonstrated that local nonlinearity and weak coupling between nonlinear split ring resonators in the chain (oneedimensional discrete system, where each split ring resonator interacts with nearest neighh bors, can lead to the formation of discrete localized structures [7, 8]. In this work, we study two types of discrete localized structures in such systems, namely, switching waves and dissipative solitons. We show that switching waves are immobile or move depending on the parameters of the system and prehistory (initial conditions), whereas solitons are highly localized due to the discreteness of the system. Following [6], we consider a periodic chain of identical nonlinear split ring resonators (see Fig. 1), which is a simple oneedimensional model of a magg netic metamaterial recently created and studied experimentally [9]. All split ring resonators lie in one plane and their centers are located on a straight line. Each split ring resonator can be associated with a nonn linear electric oscillatory circuit consisting of a nonn linear ohmic resistance, inductance, and capacitance. We assume that the nonlinearity of split ring resonators is caused by the Kerr nonlinearity of a medium introo duced into the gaps of split ring resonators [4, 6]. The master equation for the amplitude of the elecc tric current Ψ n in the nth resonator of the chain has the dimensionless form [6] (1) Here, t is the time divided by the period of natural oscillations of an isolated resonator; Ω and γ is the deviation of the eigenfrequency from the frequency of …