Motionless and moving bright dissipative solitons in an optical fiber with a Bragg grating and nonlinear amplification and absorption are found numerically and studied. These solitons form a one-parameter family whose parameter-the soliton velocity-can continuously change in a certain range. The presence of saturation of the nonlinearity is necessary for stability of such solitons. Neglect of saturation of the cubic-infield polarization of the medium results in the instability of the possible localized structures.
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