  The spatial dynamics of pulses in Kerr media with parabolic index profile are
examined. It is found that when diffraction and graded-index have opposite
signs propagating pulses exhibit an oscillatory pattern, similar to a breathing
behavior. Furthermore, if the pulse and the index profile are not aligned the
pulse oscillates around the index origin with frequency that depends on the
values of the diffraction and index of refraction. These oscillations are not
observed when diffraction and graded-index share the same sign.
