  We introduce the `Illustris zoom simulation project', which allows the study
of selected galaxies forming in the $\Lambda$CDM cosmology with a 40 times
better mass resolution than in the parent large-scale hydrodynamical Illustris
simulation. We here focus on the starburst properties of the gas in four
cosmological simulations of major mergers. The galaxies in our high-resolution
zoom runs exhibit a bursty mode of star formation with gas consumption
timescales 10 times shorter than for the normal star formation mode. The strong
bursts are only present in the simulations with the highest resolution, hinting
that a too low resolution is the reason why the original Illustris simulation
showed a dearth of starburst galaxies. Very pronounced bursts of star formation
occur in two out of four major mergers we study. The high star formation rates,
the short gas consumption timescales and the morphology of these systems
strongly resemble observed nuclear starbursts. This is the first time that a
sample of major mergers is studied through self-consistent cosmological
hydrodynamical simulations instead of using isolated galaxy models setup on a
collision course. We also study the orbits of the colliding galaxies and find
that the starbursting gas preferentially appears in head-on mergers with very
high collision velocities. Encounters with large impact parameters do typically
not lead to the formation of starbursting gas.
