Complete solution of a new superconducting current state above the critical current

A complete set of exact solutions of a new superconducting state is presented for the one-dimensional case. The order parameter $\Psi$ is periodic in space and stable in time for spatial periods long compared to the coherence length. These solutions have critical currents which are larger than the usually accepted critical current $I_c$ but with the addition of dc electric fields inside the superconductor which are generated by phase slippage of the order parameter at points where $\Psi =0\mathrm{}$. At these points the complex order parameter has a discontinuous gradient and its real amplitude a continuous gradient. Spatially periodic super and normal dc currents coexist in the superconductor simultaneously. These solutions give rise to voltage steps in the dc voltage-current characteristics for currents $I_T$ larger than $I_c$, but may in principle also exist for $I_T<I_c$ if the condition of lowest energy is waived. A critical value of a parameter $P=P_c$ is calculated below which the material cannot have superconducting states for $I_T>I_c$. $P$ is inversely proportional to the normal-state resistivity and relaxation time $\tau$ of the superconductor. $P$ can be determined from experiments and from it, $\tau$ calculated.