  We report experimental and numerical analysis of expontentially shaped long
Josephson junctions with lateral current injection. Quasi-linear flux flow
branches are observed in the current-voltage characteristic of the junctions in
the absence of magnetic field. A strongly asymmetric response to an applied
magnetic field is also exhibited by the junctions. Experimental data are found
in agreement with numerical predictions and demonstrate the existence of a
geometry-induced potential experienced by the flux quanta in nonuniform width
junctions.
