Quasi-periodic solutions of \(2k\) order wave equations (Q2910542)

This paper considers one-dimensional nonlinear \(2k\)-order wave equations NEWLINE\[NEWLINEu_{tt}+\sum{\bar r=1}^{2k}(-1)^{\bar r}\frac{\partial^{2\bar r}u}{\partial x^{2\bar r}}+mu=f(u)NEWLINE\]NEWLINE with Dirichlet boundary conditions, where the nonlinearity \(f(u)=O(u^3)\) is a real analytic odd function. The author proves that for almost all \(m>0\), the above equations admit small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori for an associate infinite-dimensional dynamical system. The proof is based on an infinite-dimensional KAM theorem and partial Birkhoff normal form techniques.