Abstract :  In this talk, I will present results on a singular perturbation problem modeling domain walls. I will discuss the existence of solutions both when the perturbation parameter is non-zero and when it is set to zero (Thomas-Fermi approximation), demonstrating their continuous connection as the parameter approaches zero. Finally, I will show that the behavior of one of the variables can be modeled by a Painlevé II equation in the limit, by the use of an appropriate change of variables.