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Applied Math Seminar
New singular solutions of the nonlinear Schrodinger equation
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The study of singular solutions of the critical nonlinear Schr\"odinger equation~(NLS) during the last twenty years has lead to the belief that the only stable singular solutions are those that collapse according to the loglog law with a self-similar profile which is the ground state of the equation $\Delta R -R + R^{4/d+1} = 0$. In this study we present numerical simulations of a new type of singular solutions of the NLS, that collapse with a self-similar ring profile (which is different from the R profile) at a square root blowup rate. We observe that the self-similar ring profile is an attractor for a large class of radially-symmetric initial conditions, but is unstable under symmetry-breaking perturbations. The equation for the ring profile admits also multi-ring solutions that give rise to collapsing self-similar multi-ring solutions, but these solutions are unstable even in the radially-symmetric case, and eventually collapse with a single ring profile. Joint work with N. Gavish and X.P. Wang |