  In order to investigate possible topological vortex structures in generalized
models, we developed a perturbative generation approach for scalar-vector
theories. We demonstrate explicitly that the dielectric permeability functions
must have a nonpolynomial shape, i. e., the form of the logarithmic function.
Basing on this result, we built models in $(2+1)D$ with logarithmic dielectric
permeability in order to investigate the presence of topological vortex
structures in a Maxwell model. This type of scalar-vector models is important
because they can generate stationary field solutions in theories describing the
dynamics of the scalar field. As examples, we chose models of the complex
scalar field coupled to the Maxwell field. Subsequently, we investigated the
model's Bogomol'nyi equations to describe the field configurations. Then, we
demonstrate numerically, for an ansatz with rotational symmetry, that the
solutions of the complex scalar field generating minimum energy configurations
are topological structures depending on the parameters obtained in the
perturbative generation of the vector-scalar theory.
