Localized-wave interactions for the discrete nonlinear Schrödinger equation under the nonvanishing background

In this paper, localized-wave interactions under the nonvanishing background are studied through the Darboux transformation (DT) for the discrete nonlinear Schrödinger equation. First of all, via the elementary DT, we obtain the first-order breather solution and give the parameter conditions for generating Kuznetsov-Ma breathers, Akhmediev breathers, breathers with a number of bunches and spatio-temporal breathers. Moreover, we analyze the effects of parameters on the velocity and period of the breathers. Secondly, we derive the second-order solution and discuss the dynamic behaviors of breather-breather and breather-rogue wave interactions on the nonvanishing background. Finally, via the generalized DT, we construct the equal-eigenvalue degenerate second-order breather solution, and analyze the characteristics of interactions between two breathers, which is found to be different from the ones given by the elementary DT.