Higher-order semirational solutions and nonlinear wave interactions for a derivative nonlinear Schrödinger equation

• The semirational solution in terms of the determinant form of the DNLS equation (The nonlinear combinations of breathers and rogue waves).
• The interaction between the breather and rogue wave.
• The formation mechanism of the higher-order rogue wave.

We present the semirational solution in terms of the determinant form for the derivative nonlinear Schrödinger equation. It describes the nonlinear combinations of breathers and rogue waves (RWs). We show here that the solution appears as a mixture of polynomials with exponential functions. The k  -order semirational solution includes k−1k−1 types of nonlinear superpositions, i.e., the l-order RW and (k-l  )-order breather for l=1,2,…,k−1l=1,2,…,k−1. By adjusting the shift and spectral parameters, we display various patterns of the semirational solutions for describing the interactions among the RWs and breathers. We find that k-order RW can be derived from a l  -order RW interacting with 12(k−l)(k+l+1) neighboring elements of a (k−lk−l)-order breather for l=1,2,…,k−1l=1,2,…,k−1.