  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. 
Classical Duals of Derivatively Self-Coupled Theories