Long time dynamics of quasi-linear Hamiltonian Klein-Gordon equations on the circle

Abstract We consider a class of Hamiltonian Klein-Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size $\eps$ showing that the corresponding solutions remain close to oscillatory motions over a time scale $\eps^{-9/4+\delta}$ for any $\delta>0$. The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.