Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient

We report approximate analytical solutions to the (3+1)-dimensional spatiotemporal nonlinear Schr\"odinger equation, with the uniform self-focusing nonlinearity and a variable negative radial diffraction coefficient, in the form of three-dimensional solitons. The model may be realized in artificial optical media, such as left-handed materials and photonic crystals, with the anomalous sign of the group-velocity dispersion (GVD). The same setting may be realized through the interplay of the self-defocusing nonlinearity, normal GVD, and positive variable diffraction. The Hartree approximation is utilized to achieve a suitable separation of variables in the model. Then, an inverse procedure is introduced, with the aim to select a suitable profile of the modulated diffraction coefficient supporting desirable soliton solutions (such as dromions, single- and multilayer rings, and multisoliton clusters). The validity of the analytical approximation and stability of the solutions is tested by means of direct simulations.