Beating solitons with nonzero relative wavenumber, their interaction, and coexistence with Akhmediev breathers

We study beating solitons and their interaction for the three-component nonlinear Schrödinger equations (NLSEs) in both focusing and defocusing cases. We identify a previously unknown family of beating solitons with nonzero relative wavenumber which is absent in the two-component NLSEs. Existence diagrams of these beating solitons are extracted from the exact solutions. Unique coexistence and interaction between beating solitons and vector Akhmediev breathers are revealed by using the existence diagrams. It is demonstrated that the coexisting breathers turn out to be the two-component Akhmediev breathers with nonzero relative wavenumber. The validity of our exact solutions has been confirmed by numerical simulations.