Self-focusing of optical pulses in media with normal dispersion

The self-focusing of ultra short optical pulses in a nonlinear medium with normal (i.e., negative) group-velocity dispersion is investigated. By using a combination of various techniques like virial-type arguments and self-similar transformations, we obtain strong evidence suggesting that a pulse propagating in a nonlinear medium with normal dispersion will not collapse to a singularity in the transverse diffraction plane. It is explicitly shown that the pulse spreads out along the time-direction and ultimately splits up. The analytical results are supported by direct numerical solutions.