  We study the local well-posedness of the initial value problem for cubic
Horndeski theories. Three different strongly hyperbolic modifications of the
ADM formulation of the Einstein equations are extended to cubic Horndeski
theories in the weak field regime. In the first one, the equations of motion
are rewritten as a coupled elliptic-hyperbolic system of partial differential
equations. The second one is based on the BSSN formulation with a generalised
Bona-Mass\'o slicing (covering the 1+log slicing) and non-dynamical shift
vector. The third one is an extension of the CCZ4 formulation with a
generalised Bona-Mass\'o slicing (also covering the 1+log slicing) and a gamma
driver shift condition. This final formulation may be particularly suitable for
applications in non-linear numerical simulations.
