  We study the polarization dynamics of ultrafast solitons in mode-locked fiber
lasers. We find that when a stable soliton is generated, it's
state-of-polarization shifts toward a stable state and when the soliton is
generated with excess power levels it experiences relaxation oscillations in
its intensity and timing. On the other hand, when a soliton is generated in an
unstable state-of-polarization, it either decays in intensity until it
disappears, or its temporal width decreases until it explodes into several
solitons and then it disappears. We also found that when two solitons are
simultaneously generated close to each other, they attract each other until
they collide and merge into a single soliton. Although, these two solitons are
generated with different states-of-polarization, they shift their
state-of-polarization closer to each other until the polarization coincides
when they collide. We support our findings by numerical calculations of a
non-Lagrangian approach by simulating the Ginzburg-Landau equation governing
the dynamics of solitons in a laser cavity. Our model also predicts the
relaxation oscillations of stable solitons and the two types of unstable
solitons observed in the experimental measurements.
