Invariant tori for the Hamiltonian derivative wave equation with higher order nonlinearity

Abstract

In this paper, we will study the Hamiltonian derivative wave equation with higher order nonlinearity \begin{document}$ y_{tt}-y_{xx}+my+(Dy)^5 = 0, \quad x\in\mathbb{T}: = \mathbb{R}/2\pi\mathbb{Z}, $\end{document} where $ m>0 $ is a potential and \begin{document}$ D: = \sqrt{-\partial_{xx}+m}. $\end{document} We will prove that, for any integer $ b\geq2 $, the above equation admits many small amplitude quasi-periodic solutions corresponding to $ b $-dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on infinite dimensional KAM theory and partial Birkhoff normal form.