DEFOCUSING REGIMES OF NONLINEAR-WAVES IN MEDIA WITH NEGATIVE DISPERSION

Defocusing regimes of quasimonochromatic waves governed by a nonlinear Schrodinger equation with mixed-sign dispersion are investigated. For a power-law nonlinearity, we show that localized solutions to this eq uation 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. Solu tions defined at the supercritical dimension are proved to exhibit a n onvanishing mean longitudinal size, and cannot transversally collapse if they are assumed to shrink along each spatial direction.