Chaotic and phase-locked breather dynamics in the damped and parametrically driven sine-Gordon equation

The dynamics of a breather in the damped and parametrically driven sine-Gordon equation is investigated both numerically and analytically. The Kahunen-Lo\`eve expansion is applied to extract the energetically dominant localized modes. These modes are used in a Galerkin approximation to the original partial differential equation. The solutions of the resulting amplitude equations are then compared to numerical simulations of the perturbed sine-Gordon equation showing a perfect agreement. In addition, two collective-coordinate models (based on a direct approach and on the inverse-scattering transform) are constructed and their limitations in comparsion with the Kahunen-Lo\`eve expansion and direct simulations are discussed. Finally, information from the periodic spectral theory and linear stability analysis is used to identify the Kahunen-Lo\`eve modes and to show why this approach gives rather good results.