Stability of Mode-Locked Kinks in the AC Driven and Damped Sine-Gordon Lattice
Kink dynamics in the underdamped and strongly discrete sine-Gordon lattice that is driven by the oscillating force is studied. The investigation is focused mostly on the properties of the mode-locked states in the overband case, when the driving frequency
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Abstract Kink dynamics in the underdamped and strongly discrete sine-Gordon lattice that is driven by the oscillating force is studied. The investigation is focused mostly on the properties of the mode-locked states in the overband case, when the driving frequency lies above the linear band. With the help of Floquet theory it is demonstrated that the destabilizing of the mode-locked state happens either through the Hopf bifurcation or through the tangential bifurcation. It is also observed that in the overband case the standing mode-locked kink state maintains its stability for the bias amplitudes that are by the order of magnitude larger than the amplitudes in the low-frequency case.
Main Abbreviations ı SG: Sine-Gordon ı FK: Frenkel-Kontorova ı JJ: Josephson junctions
ı DSG: Discrete sine-Gordon ı PN: Peierls-Nabarro ı JJA: Josephson junction arrays
1 Introduction The discrete sine-Gordon (DSG) equation, also known as the Frenkel-Kontorova (FK) model, is ubiquitous in condensed matter physics [1, 2]. It has a wide range of applications in the dislocation theory [3], weak superconductivity [4, 5] and magnetism [6]. Among the intensively discussed problems for the DSG dynamics Y. Zolotaryuk () Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, vul. Metrologichna 14-B 03680 Kiev, Ukraine 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__12, © Springer International Publishing Switzerland 2014
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the problem of the topological soliton (fluxon) response to the ac (time-periodic) bias, remains to be important. This interest is caused in particular by the number of technological applications based on the Josephson junction arrays (JJAs), which are successively modeled by the DSG equation. Properties of the small ac-biased Josephson junctions have been extensively studied both experimentally (starting from the pioneering papers of Shapiro [7]) and theoretically (with the focus on the phase-locking [8] and chaotic regimes [9,10]). In particular, the rf-biased Josephson junctions have been used as a voltage standard [10, 11]. It is well-known [12] that contrary to the continuous sine-Gordon (SG) equation the DSG equation is non-integrable, and, moreover, it does not possess moving kink solutions. The ac-driven DSG lattice has two independent sources of nonintegrability: the external drive (bias) and the discreteness. Interplay of these two sources has led to a number of interesting effects: mode-locking to the frequency of the external drive [13] and kink mobility [14, 15] (including its experimental detection in periodically modulated Josephson junctions [16]), various regimes of the dynamical chaos [13, 18], biharmonically driven discrete kink ratchet [17, 18] to name a few. However, these studies have been performed mostly in the adiabatic, subband (the driving frequency lies in the gap of the linear spectrum) or resonant (the
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