Breathers for nonlinear wave equations

Smiley, Michael W.

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 3, pp. 431-435

Résumé

The semilinear differential equation (1), (2), (3), in $\mathbb{R} \times \Omega$ with $\Omega \in \mathbb{R}^{N}$, (a nonlinear wave equation) is studied. In particular for $\Omega = \mathbb{R}^{3}$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x) = \nu(t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of Cesari.