Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential

We investigate the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with different external potentials. In the absence of external potential or in the presence of harmonic potential, the number of localized nonlinear waves is associated not only with the boundary condition but also with the singularity of inhomogeneous cubic-quintic nonlinearities; while in the presence of periodic external potential, the periodic inhomogeneous cubic-quintic nonlinearities, together with the boundary condition, support the periodic solutions with an arbitrary number of circular rings in every unit. Our results may stimulate new matter waves in high-dimensional Schrödinger equations with spatially modulated nonlinearities.