Singular standing-ring solutions of nonlinear partial differential equations

TL;DR

This paper introduces a framework for constructing singular solutions of nonlinear PDEs that blow up on a sphere, extending known point blowup solutions to higher-dimensional spherical blowup scenarios.

Contribution

It develops a general method to generate spherical blowup solutions for various nonlinear evolution equations, linking their asymptotic behavior to one-dimensional solutions.

Findings

Numerical evidence of spherical blowup solutions in multiple nonlinear PDEs.

Asymptotic profiles match those of one-dimensional singular solutions.

Blowup rates are consistent with theoretical predictions.

Abstract

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.