  Horndeski theory is the most general scalar-tensor theory retaining
second-order field equations, although the action includes higher-order terms.
This is achieved by a special choice of coupling constants. In this paper, we
investigate thick brane system in reduced Horndeski theory, especially the
effect of the non-minimal derivative coupling on thick brane. First, the
equations of motion are presented and a set of analytic background solutions
are obtained. Then, to investigate the stability of the background scalar
profile, we present a novel canonically normalized method, and show that
although the original background scalar field is unstable, the canonical one is
stable. The stability of the thick brane under tensor perturbation is also
considered. It is shown that the tachyon is absent and the graviton zero mode
can be localized on the brane. The localized graviton zero mode recovers the
four-dimensional Newtonian potential and the presence of the non-minimal
derivative coupling results in a splitting of its wave function. The correction
of the massive graviton KK modes to the Newtonian potential is also analyzed
briefly.
