Adiabatic and pendulum approximations to the retarded Josephson equation

The retarded Josephson equation, which is a second-order differential-integral equation, can be approximated by differential equations in different ways. There are the well-known adiabatic approximations which are justifiable for small ac-voltage amplitudes and in the slow-motion regime. Another type of approximation makes use of the pendulum analogy, where the arguments of the functions entering in the adiabatic current phase relation are replaced by time averages. Several modifications of this average pendulum approximation will be discussed. In particular, the characteristics of these different approximations are compared with the exact result in a limit where the agreement should be rather good. In the same limit the solutions are also investigated by asymptotic expansions under very general conditions.