Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle

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 \(\varepsilon \) $\epsilon$ showing that the corresponding solutions remain close to oscillatory motions over a time scale \(\varepsilon ^{{-\frac{9}{4}+\delta }}\) ${\epsilon }^{-\frac{9}{4}+\delta }$ for any \(\delta >0\) $\delta >0$ . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.