General solution of overdamped Josephson junction equation in the case of phase-lock

Abstract

The first order nonlinear ordinary differential equation equation ϕ˙(t) + sin ϕ(t) = B + A cos ωt, which is commonly used as a simple model of an overdamped Josephson junction in superconductors is investigated. Its general solution is obtained in the case known as phase-lock where all but one solution converge to a common ‘essentially periodic’ attractor. The general solution is represented in explicit form in terms of the Floquet solution of a double confluent Heun equation. In turn, the latter solution is represented through the Laurent series which defines an analytic function on the Riemann sphere with punctured poles. The series coefficients are given in terms of infinite products of 2 × 2 matrices with a single zero element. The closed form of the phase-lock condition is obtained and represented as the condition for existence of a real root of a transcendental function. The efficient phase-lock criterion is conjectured, and its plausibility is confirmed in numerical tests.