Breathers and breather-rogue waves on a periodic background for the derivative nonlinear Schrödinger equation

In this paper, we give the solutions on a periodic background in terms of the determinant form for the derivative nonlinear Schrödinger equation. Because its rogue wave on a periodic background has been studied, we investigate only the breather and breather-rogue wave on a periodic background for the derivative nonlinear Schrödinger equation. We obtain Kuznetsov–Ma breather, Akhmediev breather and spatio-temporal breather on a periodic background for this equation. In addition, we mainly focus on three types of the breather-rogue wave on a periodic background: (1) the interaction between a Peregrine soliton and a breather; (2) the interaction between a Peregrine soliton and two breathers; (3) the interaction between a second-order rogue wave and a breather. For the first type, we analyse the effects of the free parameters on its dynamical behaviour. The second type is described as 'rogue wave quanta' on a periodic background. The third type has two spatial-temporal distribution structures: the fundamental structure and the triangular structure.