  The flux of neutrinos from annihilation of gravitationally captured dark
matter in the Sun has significant constraints from direct-detection
experiments. However, these constraints are relaxed for inelastic dark matter
as inelastic dark matter interactions generate less energetic nuclear recoils
compared to elastic dark matter interactions. In this paper, we explore the
possibility for large volume underground neutrino experiments to detect the
neutrino flux from captured inelastic dark matter in the Sun. The neutrino
spectrum has two components: a mono-energetic "spike" from pion and kaon decays
at rest and a broad-spectrum "shoulder" from prompt primary meson decays. We
focus on detecting the shoulder neutrinos from annihilation of hadrophilic
inelastic dark matter with masses in the range 4-100 GeV and the mass
splittings in up to 300 keV. We determine the event selection criterion for
DUNE to identify GeV-scale muon neutrinos and anti-neutrinos originating from
hadrophilic dark matter annihilation in the Sun, and forecast the sensitivity
from contained events. We also map the current bounds from Super-Kamiokande and
IceCube on elastic dark matter, as well as the projected limits from
Hyper-Kamiokande, to the parameter space of inelastic dark matter. We find that
there is a region of parameter space that these neutrino experiments are more
sensitive to than the direct-detection experiments. For dark matter
annihilation to heavy-quarks, the projected sensitivity of DUNE is weaker than
current (future) Super (Hyper) Kamiokande experiments. However, for the
light-quark channel, only the spike is observable and DUNE will be the most
sensitive experiment.
