  In the present work we explore the competition of quadratic and quartic
dispersion in producing kink-like solitary waves in a model of the nonlinear
Schr{\"o}dinger type bearing cubic nonlinearity. We present the first 6
families of multikink solutions and explore their bifurcations as the strength
of the quadratic dispersion is varied. We reveal a rich bifurcation structure
for the system, connecting two-kink states with states involving 4-, as well as
6-kinks. The stability of all of these states is explored. For each family, we
discuss a ``lower branch'' adhering to the energy landscape of the 2-kink
states. We also, however, study in detail the ``upper branches'' bearing higher
numbers of kinks. In addition to computing the stationary states and analyzing
their stability within the partial differential equation model, we develop an
effective particle ordinary differential equation theory that is shown to be
surprisingly efficient in capturing the kink equilibria and normal (as well as
unstable) modes. Finally, the results of the bifurcation analysis are
corroborated by means of direct numerical simulations involving the excitation
of the states in a targeted way in order to explore their instability-induced
dynamics.
