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Discrete spectrum of kink velocities in Josephson structures: the nonlocal double sine-Gordon model

We study a model of Josephson layered structure which is characterized by two peculiarities: (i) superconducting layers are thin; (ii) due to suppression of superconducting states in superconducting layers the current-phase relation is non-sinusoidal and is described by two sine harmonics. The governing equation is a nonlocal generalization of double sine-Gordon (NDSG) equation. We argue that the dynamics of fluxons in the NDSG model is unusual. Specifically, we show that there exists a set of particular velocities for non-radiating fluxon propagation. In dynamics the presence of these ``priveleged'' velocitied results in phenomenon of quantization of fluxon velocities: in our numerical experiments a travelling kink-like excitation radiates energy and slows down to one of these particular velocities, taking a shape of predicted 2-pi-kink. This situation differs from both, double sine-Gordon local model and the nonlocal sine-Gordon model, considered before. We conjecture that the set of these velocities is infinite and present an asymptotic formula for them.