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 epsilon showing that the corresponding solutions remain close to oscillatory motions over a time scale epsilon (-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.