QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION

Abstract

In this paper, one-dimensional (1D) nonlinear wave equation $$u_{tt}-u_{xx}+mu+u^{7}=0$$on the finite $x$-interval $[0,\pi]$ with Dirichlet boundary conditions is considered. It is proved that there are many $3$-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. This is an extension of the previous work [11] by the same author, where many $2$-dimensional elliptic invariant tori for the above equation are obtained. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.