  In this paper we study a novel means of coupling neutrinos to a Lorentz
violating background k-essence field. We first look into the effect that
k-essence has on the neutrino dispersion relation and derive a general formula
for the neutrino velocity in the presence on a k-essence background. The
influence of k-essence coupling on neutrino oscillations is then considered. It
is found that a non-diagonal k-essence coupling leads to an oscillation length
that goes like \lambda \sim E^{-1} where E is the energy. This should be
contrasted with the \lambda \sim E dependence seen in the standard mass-induced
mechanism of neutrino oscillations. While such a scenario is not favored
experimentally, it places constraints on the interactions of the neutrino with
a cosmological k-essence scalar background by requiring it to be flavor
diagonal. All non-trivial physical effects discussed here require the speed of
sound to be different from the speed of light and hence are primarily a
consequence of Lorentz violation.
