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 Schrödinger equation i ψ t ( t , x ) + Δ ψ + | ψ | 2 σ ψ = 0 with σ > 2 , the biharmonic nonlinear Schrödinger equation i ψ t ( t , x ) − Δ 2 ψ + | ψ | 2 σ ψ = 0 with σ > 4 , the nonlinear heat equation ψ t ( t , x ) − Δ ψ − | ψ | 2 σ ψ = 0 with σ > 0 , and the nonlinear biharmonic heat equation ψ t ( t , x ) + Δ 2 ψ − | ψ | 2 σ ψ = 0 with σ > 0 .