Collective mode and thec-axis critical current of a Josephson-coupled superconductor at high parallel magnetic fields

Using Maxwell and time-dependent Ginzburg-Landau equations in the framework of a Lawrence-Doniach model we derive dynamical equations for the order parameter phase difference in Josephson-coupled layered superconductors. These equations allow the description of nonstationary Josephson effects in magnetic fields parallel to the layers. We obtain the dispersion relation and relaxation rate of collective modes in the absence of a magnetic field and in a strong parallel magnetic field. The dispersion with respect to the momentum in the $\mathrm{ab}$ plane in high fields is much stronger that in a single Josephson junction due to the multilayer structure. This leads to a very weak $c$-axis critical current in high magnetic fields. The relaxation rate of collective modes is found to be weak in comparison with its frequency. We discuss experimental data for the $c$-axis plasma resonance and critical current in parallel magnetic fields.