  In the present work, we consider the existence and spectral stability of
multi-pulse solitary wave solutions to a nonlinear Schr\"odinger equation with
both fourth and second order dispersion terms. We first give a criterion for
the existence of a single solitary wave solution in terms of the coefficients
of the dispersion terms, and then show that a discrete family of multi-pulse
solutions exists which is characterized by the distances between the individual
pulses. We then reduce the spectral stability problem for these multi-pulses to
computing the determinant of a matrix which is, to leading order, block
diagonal. Under an additional assumption, which can be verified numerically, we
show that all multi-pulses are spectrally unstable. For double pulses,
numerical computations are presented which are in good agreement with our
analytical results.
