In this work, the self-focusing ring originated from a rotation symmetric super-Gaussian beam (RSSGB) incident on the Nd:glass medium is described in detail, with the aid of the numerical solution to the nonlinear paraxial wave equation. For a RSSGB with initial undulations, the self-focusing ring may split after traveling certain distance inside the nonlinear medium, resulting in some high intensity filaments near the verge of the beam. For a RSSGB with a certain peak power, one can select the width and rank of the beam to suppress the formation of the self-focusing ring, so that the filaments due to the breakup of self-focusing rings may be eliminated finally.