N-fold generalized Darboux transformation and breather–rogue waves on the constant/periodic background for a generalized mixed nonlinear Schrödinger equation

In this paper, a generalized mixed nonlinear Schrödinger equation, which arises in several physical applications including fluid mechanics (for the weakly nonlinear dispersive water waves), quantum field theory and nonlinear optics, is investigated. An N-fold generalized Darboux transformation (GDT) is constructed, where N is a positive integer. Based on that N-fold GDT, we derive the higher-order rational soliton solutions with the non-vanishing background. Semirational solutions on the constant/periodic background, which are composed of the mth-order rogue wave, the (k- m- r) th-order so-called nondegenerate breather and the rth-order so-called degenerate breather, are constructed, where k= 2 , 3 , … , N, m= 1 , 2 , … , k- 1 and r= 0 , 2 , 3 , …. Breathers on the dark/bright soliton or periodic wave background are presented. Based on the semirational solutions, breather–rogue waves on the constant/periodic background are discussed analytically and graphically. Classification conditions for different types of the breather–rogue waves on the constant/periodic background are given. Triangular structure of the higher-order rogue wave on the periodic background is studied and presented.