Mechanism of dissipative soliton stabilization by nonlinear gradient terms

We present a feedback mechanism for dissipative solitons in the cubic complex Ginzburg-Landau (CGL) equation with a nonlinear gradient term. We are making use of a mechanical analog containing contributions from a potential and from a nonlinear viscous term. The feedback mechanism relies on the continuous supply of energy as well as on dissipation of the stable pulse. Our picture is corroborated by the numerical solution of the full equation. A quintic contribution is not necessary for stabilization in the presence of a suitable nonlinear gradient term as we show by using a linear stability analysis for the stationary pulses. We find that the limit of vanishing magnitude of the nonlinear gradient term is singular: For exactly vanishing nonlinear gradient terms stable pulses do not exist. This situation is qualitatively different from that found for the cubic-quintic complex Ginzburg-Landau equation with a nonlinear gradient term: In this case the limit of a vanishing nonlinear gradient contribution is completely smooth. We demonstrate that, for small magnitude of the nonlinear gradient term simple types of scaling behavior are found for the amplitude, the full width at half maximum (FWHM), the velocity, and the effective frequency of the stable pulse as a function of the magnitude of the nonlinear gradient term.