05-75 Zhenguo Liang, Jiangong You

Quasi-Periodic Solutions for 1D Nonlinear Wave Equation with a General Nonlinearity (455K, pdf) Feb 21, 05

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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.

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