On rogue wave in the Kundu-DNLS equation

• We construct the Darboux transformation of the Kundu-DNLS equation.
• The soliton solutions, positon solutions and breather solutions of the Kundu-DNLS equation are given.
• We construct rogue wave solutions which are given by using the Taylor expansion of breather solutions.
• These rogue waves constitute several patterns, such as fundamental pattern, triangular pattern, circular pattern.

In this paper, the determinant representation of the n-fold Darboux transformation (DT) of the Kundu-DNLS equation is given. Based on our analysis, the soliton solutions, positon solutions and breather solutions of the Kundu-DNLS equation are given explicitly. Further, we also construct the rogue wave solutions which are given by using the Taylor expansion of the breather solution. Particularly, these rogue wave solutions possess several free parameters. With the help of these parameters, these rogue waves constitute several patterns, such as fundamental pattern, triangular pattern, circular pattern.