  We consider an infinitely long 0-$\pi$ Josephson junction consisting of 0 and
$\pi$ regions having different critical current densities $j_{c,0}$ and
$j_{c,\pi}$. The ground state of such a junction corresponds to a spontaneosly
formed asymmetric semifluxon with tails decaying on different length scales. We
calculate the depinning current of such a fractional vortex and show that it is
different for positive and negative bias polarity. We also show that upon
application of a bias current, the fractional flux (topological charge)
associated with the vortex changes. We calculate the range of fractional flux
associated with the vortex when the bias changes from negative to positive
critical (depinning) values.
