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

Abstract

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.