Symplectic Integrators for Klein-Gordon chains - Recurring formation of localized, breather-like oscillations

Ευστράτιος-Γεώργιος Ανδρέα Ευστρατιάδης
2020
Two families of symplectic integrators to obtain the time evolution of the Klein-Gordon chain were created (in C++): integrators that split the Hamiltonian into "position" and "velocity" integrable parts (T-V decomposition) and integrators that split the Hamiltonian into "unperturbed" and "perturbation" integrable parts (mixed variable decomposition - MVD). Their speed and accuracy was compared with that of a Taylor series integrator implemented by G. Voyatzis. As initial conditions we had the
more » ... entral oscillator at a non-zero momentum and zero displacement, and every other oscillator at rest. Even though the symplectic integrators can preserve energy better than the series integrator for long-term integrations, they require small time steps, which makes the integration slower than that using the Taylor series. For very long integrations, using a symplectic integrator is favourable, because the error in energy remains bounded. During the integrator evaluation process, a strange phenomenon was observed for some initial conditions: energy spreads quickly across the oscillator chain and, after enough time, a breather-like structure is formed at the center of the chain, containing most of the initial energy. After some time, the structure breaks, but some time in the future it is formed again, and so forth. The study was then aimed at figuring out which initial conditions lead to this phenomenon. A 4th order T-V symplectic integrator was used to obtain long-term time-evolutions of Klein-Gordon chains of various lengths, with various initial conditions. It was found that, for sufficiently large chains, the phenomenon occurs in a narrow range of initial momentum of the central oscillator, which is approximately \$0.81-0.84\$. With lower initial momentum, no breather-like structure was found to form, and with higher initial momentum, most of the energy was found to never propagate away from the center of the chain. For smaller chains, the range of initial momentum in which the phenomenon occurs becomes wider. To visualize [...]