Influence of Dirichlet boundary conditions on dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation.

We investigate the influence of Dirichlet boundary conditions on various types of localized solutions of the cubic-quintic complex Ginzburg-Landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves. We find that various types of nonmoving pulses and holes can accommodate Dirichlet boundary conditions by having, for holes, two halves of a pi hole at each end of the box. Moving pulses of fixed shape as they arise for periodic boundary conditions are replaced by a nonmoving asymmetric pulse, which has half a pi hole at the end of the box in the original moving direction to guarantee that Dirichlet boundary conditions are met. Moving breathing pulses as they arise for periodic boundary conditions propagate toward one end of the container and stop moving while the breathing persists indefinitely. Finally breathing and moving holes are replaced by two (nonbreathing) half pi holes at each end of the container and one hump in the bulk.