Defocusing regimes of nonlinear waves in media with negative dispersion

Defocusing regimes of quasimonochromatic waves governed by a nonlinear Schr\"odinger equation with mixed-sign dispersion are investigated. For a power-law nonlinearity, we show that localized solutions to this equation defined at the so-called critical dimension cannot collapse in finite time in the sense that their transverse (anomalously dispersing) and longitudinal (normally dispersing) extensions never vanish. Solutions defined at the supercritical dimension are proved to exhibit a nonvanishing mean longitudinal size, and cannot transversally collapse if they are assumed to shrink along each spatial direction.