Radial solitons and modulational instability in two-dimensional Ablowitz-Ladik equation for certain applications in nonlinear optics

We probe in detail, properties of solitons and modulational instability (MI) in addition to the generation of soliton-like excitations in a discrete two-dimensional (2D) Ablowitz-Ladik (AL) equation. Using the multiple-scale and Rayleigh-Ritz variational methods, we get bright and dark radial solitons. As bright radial soliton is linked to MI, we have predicted the appearance of instability areas for the wave number belonging to [ 0 , π / 2 [ in the Brillouin zone. Through the linear stability analysis, the MI criterion of appearance of instability areas are found. Furthermore, we show that the growth rate of the amplitude of MI may be significantly influenced by the attractive nonlinearity term. Numerical simulations are performed to support our analytical analysis and an excellent agreement has been obtained.