Exact solutions of semilinear radial wave equations in \(n\) dimensions (Q1883092)

Exact solutions of the semilinear radial wave equation of the form \[ -u_{tt}+u_{rr}+\frac{n-1}ru_{r}=\pm u| u| ^{q-1},\tag{1} \] where \(q\not=1\) and \(r\) denotes the radial coordinate in \(n>1\) dimensions are considered. The group-invariant solutions arising from the reduction of the wave equation (1) to nonlinear ordinary differential equations under each of the point symmetries, time translation invariance, scaling invariance, space-time inversion are discussed in Section 2. The basis of the group foliation method considered in Section 3 is the introduction of group-resolving equations (a first-order partial differential equation system) associated with the point symmetries admitted by the equation (1). Exact solutions obtained from this method, along with a discussion of their analytical features are given in Section 4. One of the eight exact solutions to (1) given in Theorem 1 has the form \[ u=\left(\frac{k (q-1)^2}{2(q(1-n)+n+1)} \left((t+c)^2- r^2\right)\right)^{1/(1-q)} \] for \(q\not=(n+1)/(n-1)\).