Exact solutions of semilinear radial wave equations in n dimensions

TL;DR

This paper derives exact solutions for n-dimensional radial wave equations with various nonlinearities using symmetry group methods, revealing solutions with blow-up behavior and static monopoles.

Contribution

It introduces a novel ansatz technique to find both group-invariant and more general exact solutions for nonlinear PDEs, expanding the solution space.

Findings

Derived exact solutions with blow-up behavior

Obtained static monopole solutions

Applied symmetry group methods to nonlinear PDEs

Abstract

Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system of group-invariant variables given by group foliations of the wave equation, using the one-dimensional admitted point symmetry groups. (These groups comprise scalings and time translations, admitted for any nonlinearity power, in addition to space-time inversions admitted for a particular conformal nonlinearity power). This is shown to yield not only group-invariant solutions as derived by standard symmetry reduction, but also other exact solutions of a more general form. In particular, solutions with interesting analytical behavior connected with blow ups as well as static monopoles are obtained.