Dynamics of the higher-order rogue waves for a generalized mixed nonlinear Schrödinger model

• Modulational instability characteristics of the generalized mixed nonlinear Schrödinger equation.
• Higher-order rogue waves in terms of the determinants for the generalized mixed nonlinear Schrödinger equation.
• The semirational rogue–wave solutions which are a combination of rational and exponential functions.
• Effects of the nonlinear parameters on the rogue waves.

Under investigation in this paper is a generalized mixed nonlinear Schrödinger equation (GMNLSE) which arises in several physical areas including the quantum field theory, weakly nonlinear dispersive water waves, and nonlinear optics. The linear stability analysis is performed and the instability zones as well as the modulational instability gain are obtained and discussed. Higher–order rogue waves (RWs) in terms of the determinants for the GMNLSE model are constructed by the N-fold Darboux transformation. Several patterns of the RWs are illustrated, such as the fundamental pattern, triangular pattern, circular pattern, pentagon pattern, circular–triangular pattern, and circular-fundamental pattern. Effects of the nonlinear parameters on the RWs are discussed. It is found that the nonlinear terms affect the widths and velocities of the RWs, although the amplitudes of these waves remain unchanged. The semirational RW solution, which is a combination of rational and exponential functions, is derived to describe the interaction between the RW and multi-breather.