Variations of $\alpha$ and $G$ from nonlinear multidimensional gravity
  To explain the recently reported large-scale spatial variations of the fine
structure constant $\alpha$, we apply some models of curvature-nonlinear
multidimensional gravity. Under the reasonable assumption of slow changes of
all quantities as compared with the Planck scale, the original theory reduces
to a multi-scalar field theory in four dimensions. On this basis, we consider
different variants of isotropic cosmological models in both Einstein and Jordan
conformal frames. One of the models turns out to be equally viable in both
frames, but in the Jordan frame the model predicts simultaneous variations of
$\alpha$ and the gravitational constant $G$, equal in magnitude. Large-scale
small inhomogeneous perturbations of these models allow for explaining the
observed spatial distribution of $\alpha$ values.
