Maxwell-scalar system and scalar field with impurity: novel approach to find nontrivial solutions

In this work, we investigate a Maxwell-scalar model that couples the scalar and gauge fields through the electric permittivity and another model, in which the scalar field lives in the presence of impurity. By considering a single spatial dimension, we determine the conditions under which the model with impurity can be seen as an effective model for the Maxwell-scalar system, having similar solutions. This correspondence shows that the impurity can be used to describe the presence of a charge density, and we use it to verify that the impurity-free case, which supports minimum energy configurations, is related to the case of point charges. We also investigate a class of impurities which modifies the core of the scalar field, and find the corresponding nontrivial charge densities and electric fields. In particular, the asymptotic behavior in terms of the impurity is also studied, leading to solutions exhibiting long-range or quasi-compact profile.