  We numerically investigate the generation of solitons in current-biased long
Josephson junctions in relation to the superconducting lifetime and the voltage
drop across the device. The dynamics of the junction is modelled with a
sine-Gordon equation driven by an oscillating field and subject to an external
non-Gaussian noise. A wide range of $\alpha$-stable L\'evy distributions is
considered as noise source, with varying stability index $\alpha$ and asymmetry
parameter $\beta$. In junctions longer than a critical length, the mean
switching time (MST) from superconductive to the resistive state assumes a
values independent of the device length. Here, we demonstrate that such a value
is directly related to the mean density of solitons which move into or from the
washboard potential minimum corresponding to the initial superconductive state.
Moreover, we observe: (i) a connection between the total mean soliton density
and the mean potential difference across the junction; (ii) an inverse behavior
of the mean voltage in comparison with the MST, with varying the junction
length; (iii) evidences of non-monotonic behaviors, such as stochastic resonant
activation and noise enhanced stability, of MST versus the driving frequency
and noise intensity for different values of $\alpha$ and $\beta$; (iv) finally,
these non-monotonic behaviors are found to be related to the mean density of
solitons formed along the junction.
