Dissipation and coherent effects in narrow superconducting channels

We apply the time dependent Ginzburg-Landau equations (TDGL) to study small ac currents of frequency $\omega$ in superconducting channels narrow on the scale of London penetration depth. We show that TDGL have $t$-dependent and spatially uniform solutions that describe the order parameter with an oscillating part of the double frequency coexisting with an ac electric field. We evaluate the Ohmic losses (related neither to the flux flow nor to the phase slips) and show that the resistivity reduction on cooling through the critical temperature $T_c$ should behave as $(T_c-T)^2/\omega^2$. If the channel is cut out of an anisotropic material in a direction other than the principal axes, the transverse phase difference and the Josephson voltage between the channel sides are generated.