The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrӧdinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.
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 "л"-shape non-autonomous discrete bright soliton solution(NDBSS), the novel "λ"-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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.