The 3D solitons and vortices in 3D discrete monatomic lattices with cubic and quartic nonlinearity

By virtue of the method of multiple-scale and the quasi-discreteness approach, we have discussed the nonlinear vibration equation of a 3D discrete monatomic lattice with its nearest-neighbours interaction. The 3D simple cubic lattices have the same localized modes as a 1D discrete monatomic chain with cubic and quartic nonlinearity. The nonlinear vibration in the 3D simple cubic lattice has 3D distorted solitons and 3D envelop solitons in the direction of k_x=k_y=k_z=k and k=\pm \pi/6a_0 in the Brillouin zone, as well as has 3D vortices in the direction of k_x=k_y=k_z=k and k=\pm \pi/a_0 in the Brillouin zone.