Time-periodic solitons in a damped-driven nonlinear Schr'odinger equation

摘要

Time-periodic solitons of the parametrically driven damped nonlinearSchr\"odinger equation are obtained as solutions of the boundary-value problemon a two-dimensional spatiotemporal domain. We follow the transformation of theperiodic solitons as the strength of the driver is varied. The resultingbifurcation diagrams provide a natural explanation for the overall form anddetails of the attractor chart compiled previously via direct numericalsimulations. In particular, the diagrams confirm the occurrence of theperiod-doubling transition to temporal chaos for small values of dissipationand the absence of such transitions for larger dampings. This difference in thesoliton's response to the increasing driving strength can be traced to thedifference in the radiation frequencies in the two cases. Finally, we relatethe soliton's temporal chaos to the homoclinic bifurcation.