 0575 Zhenguo Liang, Jiangong You
 QuasiPeriodic Solutions for 1D
Nonlinear Wave Equation with a General Nonlinearity
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Feb 21, 05

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Abstract. In this paper, onedimensional ($1D$) 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 smallamplitude quasiperiodic 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.
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