Soliton interactions and degenerate soliton complexes for the focusing nonlinear Schrödinger equation with nonzero background

We characterize soliton interactions in focusing media described by the nonlinear Schrodinger equation in the presenze of a nonzero background field, including the cases of bound states (degenerate soliton trains) and interactions between solitons and Akhmediev breathers. We first characterize bound states, which, as in the case of zero background, are obtained when several solitons travel with the same velocity. We then turn to the case when the soliton velocities are distinct, and we compute the long-time asymptotic behavior of soliton interactions by calculating the position shift for each soliton as \(t\rightarrow\pm\infty\). We also identify conditions that give rise to large position shifts. Moreover, we characterize the asymptotic phase of the nonzero background in each sector of the xt -plane that is separated by individual solitons or breathers, and we show that the asymptotic phase can be easily determined from whether the region is on the left or on the right of a soliton or an Akhmediev breather.