  We apply the Monte Carlo Wave Packet (MCWP) approach to investigate the
kinetic energy release (KER) spectra of the protons following double ionization
in H$_{2}$ when interacting with laser pulses with central wavelengths ranging
from the near-IR (800 nm) to the mid-IR (6400 nm) regions and with durations of
3-21 laser cycles. We uncover the physical origins of the peaks in the nuclear
KER spectra and ascribe them to mechanisms such as ionization following a
resonant dipole transition, charge-resonance-enhanced ionization (CREI) and
ionization in the dissociative limit of large internuclear distances. For
relatively large pulse durations, i.e., for 15 or more laser cycles at 3200 nm
and 10 or more at 6400 nm, it is possible for the nuclear wave packet in
H$_{2}^{+}$ to reach very large separations. Ionization of this part of the
wave packet results in peaks in the KER spectra with very low energies. These
peaks give direct information about the dissociative energy in the
$2p\sigma_{u}$ potential energy curve of H$_{2}^{+}$ at the one- and
three-photon resonances between the $2p\sigma_{u}$ and $1s\sigma_{g}$ curves in
H$_{2}^{+}$. With the MCWP approach, we perform a trajectory analysis of the
contributions to the KER peaks and identify the dominant ionization pathways.
Finally, we consider a pump-probe scheme by applying two delayed pulses to
track the nuclear dynamics in a time-resolved setting. Low-energy peaks appear
for large delays and these are used to obtain the $2p\sigma_{u}$ dissociative
energy values at the one-photon resonance between the $2p\sigma_{u}$ and
$1s\sigma_{g}$ curves in H$_{2}^{+}$ for different wavelengths.
