The evolution of polarized light in a nematic liquid crystal whose directors have a local response to reorienta-tion by the light is analyzed for arbitrary input light power. Approximate equations describing this evolution are derived based on a suitable trial function in a Lagrangian formulation of the basic equations governing the electric fields involved. It is shown that the nonlinearity of the material response is responsible for the forma-tion of solitons, so-called nematicons, by saturating the nonlinearity of the governing nonlinear Schrödinger equation. Therefore in the local material response limit, solitons are formed due to the nonlinear saturation behavior. It is finally shown that the solutions of the derived approximate equations for nematicon evolution are in excellent agreement with numerical solutions of the full nematicon equations in the local regime.