Regularization and control of self-focusing in the 2D cubic Schrodinger equation by attractive linear potentials

The self-focusing singularity of the attractive 2D cubic Schrödinger equation arises in nonlinear optics and many other situations, including certain models of Bose-Einstein condensates. This 2D case is very sensitive to perturbations of the equation and so solutions can be regularized in a number of ways. Here the effect of linear potentials is considered, such as could arise in models of optical fibres with narrow cores of different refractive index, wave-guides induced in a nonlinear medium by another beam, and as part of the Gross-Pitaevskii model of Bose-Einstein condensates. It is observed that in critical dimension only, one can have inhibition of collapse by attractive linear potentials, without dissipation, and that this can lead to a stable oscillating beam, as opposed to the dispersion or dissipation seen with previously studied regularizing mechanisms.