Fractional Josephson vortices at YBa$_2$Cu$_3$O$_{7-x}$ grain boundaries

We report numerical simulations of magnetic flux patterns in asymmetric 45$^{circ}$ [001]-tilt grain boundaries in YBa$_2$Cu$_3$O$_{7-x}$ superconducting films. The grain boundaries are treated as Josephson junctions with the critical current density $j_c(x)$ alternating along the junctions. We demonstrate the existence of Josephson vortices with fractional flux quanta for both periodic and random $j_c(x)$. A method is proposed to extract fractional vortices from experimental flux patterns.