  We investigate analytically long Josephson junctions with phase
pi-discontinuity points. Such junctions are usually fabricated as a ramp
between a superconductor like YBCO with d-wave symmetry of the order parameter
and an s-wave superconductor like Nb. From the top, they look like zigzags with
pi-jumps of the Josephson phase at the corners. These pi-jumps, at certain
conditions, lead to the formation of half-integer flux quanta, which we call
semi-fluxons, pinned at the corners. We derive a version of sine-Gordon
equation which describes the dynamics of the Josephson phase in such
structures, and obtain an explicit formula which describes the shape of a
semi-fluxon. Some properties of semi-fluxons are discussed. We propose a way to
construct artificial pi-junctions using only s-wave superconductors.
