Accessible solitons in complex Ginzburg-Landau media

We construct dissipative spatial solitons in one- and two-dimensional (1D and 2D) complex Ginzburg-Landau (CGL) equations with spatially uniform linear gain; fully nonlocal complex nonlinearity, which is proportional to the integral power of the field times the harmonic-oscillator (HO) potential, similar to the model of "accessible solitons;" and a diffusion term. This CGL equation is a truly nonlinear one, unlike its actually linear counterpart for the accessible solitons. It supports dissipative spatial solitons, which are found in a semiexplicit analytical form, and their stability is studied semianalytically, too, by means of the Routh-Hurwitz criterion. The stability requires the presence of both the nonlocal nonlinear loss and diffusion. The results are verified by direct simulations of the nonlocal CGL equation. Unstable solitons spontaneously spread out into fuzzy modes, which remain loosely localized in the effective complex HO potential. In a narrow zone close to the instability boundary, both 1D and 2D solitons may split into robust fragmented structures, which correspond to excited modes of the 1D and 2D HOs in the complex potentials. The 1D solitons, if shifted off the center or kicked, feature persistent swinging motion.