One- and two-collective variable descriptions of two interacting sine-Gordon kinks

Abstract

Using an ansatz [T. Matsuda, Lett. Nuovo Cimento 24 (1979) 207] which is an exact collective variable (CV) solution for kink-kink (K-K) and kink-antikink (K-A) interactions as well as for the breather in the unperturbed sine-Gordon (SG) equation and uses only one CV r( t), the separation between the kinks, we study the effects of microshort and microresistor perturbations in a Josephson junction transmission line on K-K interactions. Representing the energy stored in microshorts and microresistors, these perturbations can profoundly alter the motion of fluxons, which we term kinks, in Josephson junction transmission lines. With this ansatz, we describe particle characteristics, including relativistic effects, of K-K and K-A interactions and the breather in the unperturbed SG system. Because, in a perturbed SG system, this ansatz is no longer exact, we modify it by treating the slope of the kinks Γ( t) as a CV. We derive the masses of the r and Γ CVs, the interaction potentials, and the Hamiltonians for K-K and K-A. The inclusion of Γ( t) qualitatively and quantitatively changes the results of the one-CV r( t) description. Significantly, moderately strong microshort (microresistor) perturbations and highly relativistic initial velocities increase (decrease) the time advance, τ, defined as δ ν , where ν is the single kink velocity and δ is the kinks' increased separation or phase shift due to their interaction, in two-CV K-K scattering relative to the unperturbed τ, by almost a factor of two for the microshort, whereas these conditions produce the reverse qualitative and a smaller quantitative effect on the one-CV K-K τ. In contrast, for weaker perturbations and slower initial velocities, a microshort (microresistor) decreases (increases) the τ compared to the unperturbed τ for both the one- and two-CV descriptions.