Chaotic transport in deterministic sine-Gordon soliton ratchets

We investigate homogeneous and inhomogeneous sine-Gordon ratchet systems in which a temporal symmetry and the spatial symmetry, respectively, are broken. We demonstrate that in the inhomogeneous systems with ac driving the soliton dynamics is chaotic in certain parameter regions, although the soliton motion is unidirectional. This is qualitatively explained by a one-collective-coordinate theory which yields an equation of motion for the soliton that is identical to the equation of motion for a single particle ratchet which is known to exhibit chaotic transport in its underdamped regime. For a quantitative comparison with our simulations we use a two-collective-coordinate (2CC) theory. In contrast to this, homogeneous sine-Gordon ratchets with biharmonic driving, which breaks a temporal shift symmetry, do not exhibit chaos. This is explained by a 2CC theory which yields two ODEs: one is linear, the other one describes a parametrically driven oscillator which does not exhibit chaos. The latter ODE can be solved by a perturbation theory which yields a hierarchy of linear equations that can be solved exactly order by order. The results agree very well with the simulations.