  The general objective of the work is to study dynamics of dissipative
solitons in the framework of a one-dimensional complex Ginzburg-Landau equation
(CGLE) of a fractional order. To estimate the shape of solitons in fractional
models, we first develop the variational approximation for solitons of the
fractional nonlinear Schrodinger equation (NLSE), and an analytical
approximation for exponentially decaying tails of the solitons. Proceeding to
numerical consideration of solitons in fractional CGLE, we study, in necessary
detail, effects of the respective Levy index (LI) on the solitons' dynamics. In
particular, dependence of stability domains in the model's parameter space on
the LI is identified. Pairs of in-phase dissipative solitons merge into single
pulses, with the respective merger distance also determined by LI.
