Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

Abstract

In this paper, one dimensional nonlinear wave equation \begin{document}$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $\end{document} with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.