  The kinematics of galaxies provide valuable insights in their physics and
assembly history. Kinematics are governed not only by the gravitational
potential, but also by merger events and stellar feedback processes such as
stellar winds and supernova explosions. Aims. We aim at identifying what
governs the kinematics in a sample of SDSS selected nearby starburst galaxies,
by obtaining spatially resolved measurements of the gas and stellar kinematics.
We obtain near-infrared integral field K-band spectroscopy with VLT/SINFONI of
15 compact starburst galaxies. We derive the integrated as well as spatially
resolved stellar and gas kinematics. The stellar kinematics are derived from
the CO absorption bands, and Pa$\alpha$ and Br$\gamma$ emission lines are used
for the gas kinematics. Based on the integrated spectra we find that the
majority of galaxies have gas and stellar velocity dispersion that are
comparable. A spatially resolved comparison shows that the six galaxies that
deviate show evidence for a bulge or stellar feedback. Two galaxies are
identified as mergers based on their double peaked emission lines. In our
sample, we find a negative correlation between the ratio of the rotational
velocity over the velocity dispersion (vrot/$\sigma$) and the star formation
rate surface density. We propose a scenario where the global kinematics of the
galaxies are determined by gravitational instabilities that affect both the
stars and gas. This process could be driven by mergers or accretion events.
Effects of stellar feedback on the ionised gas are more localised and detected
only in the spatially resolved analysis. The mass derived from the velocity
dispersion provides a reliable mass even if the galaxy cannot be spatially
resolved. The technique used in this paper is applicable to galaxies at low and
high redshift with the next generation of infrared focussed telescopes (JWST,
ELT).
