Zhenguo Liang, Jiangong You
Quasi-Periodic Solutions for 1D
Nonlinear Wave Equation with a General Nonlinearity
(455K, pdf)
ABSTRACT. In this paper, one--dimensional ($1D$) wave equation with a general nonlinearity
$$
u_{tt} -u_{xx} +m u+f(u)=0,\ m>0
$$
under Dirichlet boundary conditions is considered; the
nonlinearity $f$ is a real analytic, odd function and
$f(u)=au^{2\bar{r}+1}+\sum\limits_{k\geq \bar{r}+1}f_{2k+1}u^{2k+1},\ a\neq 0\ {\rm and}\ \bar{r}\in \N$.
It is proved that for almost all $m>0$ in Lebesgue measure sense,
the above equation admits small-amplitude quasi-periodic solutions corresponding to finite
dimensional invariant tori of an associated infinite dimensional dynamical system.
The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills.