Envelope solitons of a discrete NLSE via the multi-scale quasi-discrete approximation method

The goal of this work is to obtain some envelope solitary solutions of a discrete nonlinear Schrödinger equation (NLSE) in a local optical lattice potential well through symbolic computation. By multiple scales combined with a quasi-discrete approximation method, an envelope soliton solution is constructed for the proposed equations. Moreover, the dynamics of the resulting envelope solitonary solutions are discussed. It was found that stability appeared in the system. In addition to the fixed symmetric envelope solitons, a new nonlinear element excitation, periodic kink bright and dark envelope solitons, are also observed. The degree can be controlled by the lattice constant and the depth of the optical lattice well. Consequently, it may provide a theoretical basis for the fabrication of the controllable matter-wave soliton controller and splitter.