The study of singular solutions of the NLS goes back to the 1960s, with applications in nonlinear optics and, more recently, in BEC. Asymptotic and numerical studies conducted in the 80s showed that singular solutions of the critical NLS collapse with the Townes (R) profile at a blowup rate known as the loglog law. Recently (2003) Merle and Raphael proved this result rigorously for a large class of initial conditions. Concurrently, it was demonstrated experimentally that the profile of collapsing laser beams is given by the Townes profile. Thus, all the research that was carried out from the eighties until these days leads to the belief that the Townes profile is the only attractor of blowup solutions of the critical NLS.
In this talk I will present new families of singular solutions of the critical and supercritical NLS that collapse with a self-similar ring profile, and whose blowup rate is different from the one of the "old" singular solutions. I will show, experimentally and theoretically, that these new blowup profiles are attractors for large super-Gaussian initial conditions.
In addition, I will present in the talk a semi-static adaptive grid method we have developed for the numerical simulations involved in this study for solutions which focus over 15 orders of magnitude.