Multiple-peak and multiple-ring solitons in the nonlinear Schrödinger equation with inhomogeneous self-defocusing nonlinearity

We prove that inhomogeneous defocusing cubic (Kerr) nonlinear media described by the nonlinear Schrödinger equation, which could be realized in the experiments of Bose–Einstein condensates or nonlinear optics, can support various types of one-dimensional (1D) multiple-peak and two-dimensional (2D) multiple-ring solitons, both with equal intensity peaks. The profiles of such equal-peak structures are determined by the parameters describing nonlinearity, and their relationship is clearly presented. Interestingly, the number of 1D equal peaks can be any positive odd natural number, and one of the 2D equal-annular-peak rings can be arbitrary positive integer, as long as the parameters of nonlinearity are set appropriately. It should be mentioned that the expressions on how to calculate the number of equal peaks are found phenomenologically but are clearly displayed. Besides fundamental modes, such nonlinear media can also support 2D vortical modes whose stability and instability domains appear alternately and are verified by the linear stability analysis and direct numerical simulations, which is in contrast to their fundamental nonvortical counterparts that are completely stable.