  In the study of non-equilibrium polariton condensates it is usually assumed
that the dispersion relation of polaritons is parabolic in nature. We show that
considering the true non-parabolic kinetic energy of polaritons leads to
significant changes in the behaviour of the condensate due to the curvature of
the dispersion relation and the possibility of transfer of energy to high
wavenumber components in the condensate spatial profile. We present explicit
solutions for plane waves and linear excitations, and identify the differences
in the theoretical predictions between the parabolic and non-parabolic
mean-field models, showing the possibility of symmetry breaking in the latter.
We then consider the evolution of wavepackets and show that self-localisation
effects may be observed due to the curvature of the dispersion relation.
Finally, we revisit the dynamics of dark soliton trains and show that
additional localized density excitations may emerge in the dynamics due to the
excitation of high frequency components, mimicking the appearance of
near-bright solitary waves over short timescales.
