Dynamics of excitations in Josephson media

By the “Josephson medium”, we understand a two-dimensional (2D) medium, in which currents are governed by a Josephson-type rule. Vortices are the generic excitations in such a medium. The formal description of the medium poses some problems since Lagrangian and Hamiltonian approaches reveal some discrepancies, which can be avoided if more general equations are considered. The discrepancies are natural consequences of the Poisson structure associated with Josephson medium and the choice of canonical variables. From the practical point of view, the problem can be eliminated either by reducing the number of variables or by considering a more general manifold. Taking into account a cloud of non-interacting charged particles it leads to Josephson–Vlasov equations.