On the existence of smooth breathers for nonlinear wave equations (Q1188250)

Time-periodic solutions of the nonlinear wave equation \[ u_{tt} - \Delta u+ \alpha g(u) = f,\quad (t,x) \in \mathbb{R} \times \mathbb{R}^ n \] subject to radial symmetry and decay to zero at infinity are calculated numerically. The numerical procedure is based on the method of `alternative problems', which is closely related to Lyapunov-Schmidt reduction. In one space dimension these breathers are calculated for a variety of nonlinear functions \(g(u)\) and for different forcing functions \(f(t,x)\). In three space dimensions mainly the radially symmetric sine- Gordon equation \(g(u) = \sin(u)\) is considered. In this case an extra regularity condition at the origin constitutes the missing bifurcation equation. The numerical results reported in this paper should assist the analysis of this type of problems.