Stability of Localized Periodic Orbits in the FPUT Model

The FPUT model is a classical system with weak integrability breaking, as it can be viewed as a deviation from either a chain of linearly coupled oscillators or from the integrable Toda lattice. Periodic orbits known as q-breathers can be continued from the normal modes of the linear chain to finite nonlinearity, where they remain localized in mode space. The q-breathers resemble solitons in the sense that their energy density is also localized in real space, with oscillator displacements forming a kink-like profile which propagates with energy dependent speed. We numerically study the stability of q-breathers over a range of nonlinear couplings through the Floquet matrix, which gives a linear mapping of the time evolution of infinitesimal perturbations over the q-breather period. We find that the eigenvalues of the Floquet matrix generally lie on the unit circle, but can leave at particular values of the nonlinear coupling where two or more eigenvalues collide, leading to instability. We study the effect of these instabilities on trajectories lying close to a q-breather, in particular for states prepared approximately adiabatically from a linear mode.