Shock waves in superconducting cosmic strings: growth of current
  Intrinsic equations of motion of superconducting cosmic string may admit solutions in the shock-wave form that implies discontinuity of the current term \chi. The hypersurface of discontinuity propagates at finite velocity determined by finite increment \Delta \chi =\chi_+ -\chi_-. The current increases \chi_+>\chi_- in stable shocks but transition between spacelike (\chi >0) and timelike (\chi<0) currents is impossible. 
Ernst Trojan and George V. Vlasov
hep-th astro-ph.HE hep-ph