We study the polarization dynamics of ultrafast solitons in mode-locked fiber lasers. We found that when a stable soliton is generated, its 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. All our results are supported by both experimental measurements and calculated results. For numerically modeling the dynamics of ultrafast solitons we resort to a non-Lagrangian approach for simulating coupled complex Ginzburg-Landau equations for the two components of the electric wave vector. Here we present the numerical code and results and explain in details how we obtained them.
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