Dynamics of dark and anti-dark solitons for the x-nonlocal Davey–Stewartson II equation

We investigate the x-nonlocal Davey–Stewartson II equation based on Kadomtsev–Petviashvili hierarchy reduction method, and then report dark solitons and (semi-) rational solutions expressed in the Gram-type determinant. As an application of those obtained analytical solutions, we study the evolution scenarios of the dark/anti-dark solitons on nonzero backgrounds. In addition, we analyze three kinds of the elastic interactions between the dark solitons and/or anti-dark solitons via the asymptotic analysis. In particular, we present the discovery of degenerate two solitons as single dark soliton or single anti-dark soliton. Besides, we investigate five kinds of the four solitons and four kinds of the degenerate four solitons. We find that the degenerate four solitons are different from the general three solitons, since the invisible soliton will still affect the three visible solitons in the interaction region.