Degeneracy in bright–dark solitons of the Derivative Nonlinear Schrödinger equation

Abstract

Bright–dark soliton interactions modelled by the Derivative Nonlinear Schrödinger (DNLS) equation are constructed from non-vanishing boundary conditions by the N-fold Darboux transformation. Adjusting the limitation λk→λc1(λc2), where λc1(λc2) is a critical eigenvalue associated with the synchronization of the relative phase of the bright–dark solitons in the interacting area, enables to obtain different types of quasi-rational solutions from the bright–dark solitons degeneration. Namely, quasi-rational bright and dark solitons, quasi-rational bright–dark solitons and rogue waves are found. Since a large number of preceding researchers have already addressed the relationship between the breather solutions and rogue waves, a more general modelling of rogue waves can be realized via the consideration of degeneration of bright–dark solitons. Hence, the phenomenon discussed here represents a novel nonlinear mechanism for the generation of rogue waves.