Localized nonlinear wavepackets with radial-azimuthal modulated nonlinearity and an external potential

We analytically investigated two-dimensional localized nonlinear waves in Kerr media with radial and azimuthal modulation of the nonlinearity and in the presence of an external potential. The solutions have been derived through the similarity transformation. We demonstrate that the properties of nonlinear waves are determined by two parameters: a whole number n (the index of the Jacobi elliptical waves) and an integer m (the topological charge). Our results indicate that the dynamic evolution, including cnoidal and snoidal waves, can be strongly affected by these two parameters, providing an approach to controlling nonlinear waves by an appropriate radial-azimuthal modulation of the nonlinearity, with an appropriate external potential.