  We construct the non uniform AdS black string solution with a perturbation
theory in six dimensions, focusing on the backreacting second order correction.
The backreactions at second order give the first relevant corrections to the
thermodynamical quantities. Our results show that for configurations with
horizon radius and length in the extradimension small compared to the AdS
radius, the properties of the non uniform black string are similar to the
locally asymptotically flat case. For black strings with small horizon radial
coordinate but large length in the extradimension, the thermodynamical
properties of the solutions are affected by the AdS curvature.
