Interaction dynamics of nonautonomous bright and dark solitons of the discrete (2 + 1)-dimensional Ablowitz–Ladik equation

The non-autonomous discrete bright–dark soliton solutions(NDBDSSs) of the 2 + 1-dimensional Ablowitz–Ladik (AL) equation are derived. We analyze the dynamic behaviors and interactions of the obtained 2 + 1-dimensional NDBDSSs. In this paper, we present two kinds of different methods to control the 2 + 1-dimensional NDBDSSs. In first method, we can only control the wave propagations through the spatial part, since the time function has not effect in the phase part. In second method, we can control the wave propagations through both the spatial and temporal parts. The different propagation phenomena can also be produced through two kinds of managements. We obtain the novel “ $$\pi $$ ”-shape non-autonomous discrete bright soliton solution(NDBSS), the novel “ $$\curlywedge $$ ”-shape non-autonomous discrete dark soliton solution(NDDSS) and their interaction behaviors. The novel behaviors are considered analytically, which can be applied to the electrical and optical fields.