Next-next-nearest neighbor hopping effects on the MI and on the dynamics of breathers in 2D quantum ultracold atoms loaded in optical lattices

We explore analytically and numerically the dual role played by the next-next-nearest neighbor (NNNN) hopping coupling on the modulation instability (MI) and on the dynamics of breathers in 2D quantum ultracold atoms loaded in optical lattices. Via the linear stability analysis, it is shown that the dispersion relation formed exhibits intriguing forms. It is found that the emergence of MI regions and the growth rate may be significantly affected by the NNNN hopping coupling strength. To support the analytical studies, direct numerical simulations of MI are carried out to show the generation of a train of short waves exhibiting periodic W-shaped and V-shaped solitons with decreasing amplitude as time evolves. The appearance of breathers in the regions where the MI manifests is predicted to be influenced by the NNNN hopping coupling strength. By making use of Rayleigh-Ritz variational approach and in agreement with the MI analysis, the analytical results reveal the existence of the radial modes and discrete vortex solitons in called this framework, the dynamics of breathers. The accuracy of the outcomes is checked by numerical calculations which show a good agreement with the theoretical analysis.