  Solutions to scalar theories with derivative self-couplings often have
regions where non-linearities are important. Given a classical source, there is
usually a region, demarcated by the Vainshtein radius, inside of which the
classical non-linearities are dominant, while quantum effects are still
negligible. If perturbation theory is used to find such solutions, the
expansion generally breaks down as the Vainshtein radius is approached from the
outside. Here we show that it is possible, by integrating in certain auxiliary
fields, to reformulate these theories in such a way that non-linearities become
small inside the Vainshtein radius, and large outside it. This provides a
complementary, or classically dual, description of the same theory -- one in
which non-perturbative regions become accessible perturbatively. We consider a
few examples of classical solutions with various symmetries, and find that in
all the cases the dual formulation makes it rather simple to study regimes in
which the original perturbation theory fails to work. As an illustration, we
reproduce by perturbative calculations some of the already known
non-perturbative results, for a point-like source, cosmic string, and domain
wall, and derive a new one. The dual formulation may be useful for developing
the PPN formalism in the theories of modified gravity that give rise to such
scalar theories.
