Discrete set of kink velocities in Josephson structures: The nonlocal double sine–Gordon model

• We study a nonlocal model which describes a Josephson layered structure.
• There exists a discrete set of velocities for non-radiating fluxon propagation.
• Numerical modeling shows that these velocities appear spontaneously in dynamics.
• An asymptotical formula for these velocities is conjectured.

We study a model of Josephson layered structure which is characterized by two peculiarities: (i) superconducting layers are thin; (ii) 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 constant velocities (called “sliding” velocities) for non-radiating stationary fluxon propagation. In dynamics, the presence of this set results in quantization of fluxon velocities: in numerical experiments a traveling kink-like excitation radiates energy and slows down to one of these particular constant velocities, taking the shape of predicted 2π2π-kink. We conjecture that the set of these stationary velocities is infinite and present an asymptotic formula for them.