Singular standing-ring solutions of nonlinear partial differential equations

We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrodinger equation, the biharmonic nonlinear Schrodinger equation, the nonlinear heat equation and the nonlinear biharmonic heat equation.