  The aim of this paper is to explore D-dimensional theories of pure gravity
whose space of solutions contains certain class of AdS-waves, including in
particular Schrodinger invariant spacetimes. This amounts to consider higher
order theories, and the natural case to start with is to analyze generic
square-curvature corrections to Einstein-Hilbert action. In this case, the
Schrodinger invariant sector in the space of solutions arises for a special
relation between the coupling constants appearing in the action. On the other
hand, besides the Schrodinger invariant configurations, logarithmic branches
similar to those of the so-called Log-gravity are also shown to emerge for
another special choice of the coupling constants. These Log solutions can be
interpreted as the superposition of the massless mode of General Relativity and
two scalar modes that saturate the Breitenlohner-Freedman bound (BF) of the AdS
space on which they propagate. These solutions are higher-dimensional analogues
of those appearing in three-dimensional massive gravities with relaxed AdS_3
asymptotic. Other sectors of the space of solutions of higher-curvature
theories correspond to oscillatory configurations, which happen to be below the
BF bound. Also, there is a fully degenerated sector, for which any wave profile
is admitted. We comment on the relation between this degeneracy and the
non-renormalization of the dynamical exponent of the Schrodinger spaces. Our
analysis also includes more general gravitational actions with non-polynomial
corrections consisting of arbitrary functions of the square-curvature
invariants. The same sectors of solutions are shown to exist for this more
general family of theories. We finally consider the Chern-Simons modified
gravity in four dimensions, for which we derive both the Schrodinger invariant
as well as the logarithmic sectors.
