Higher-order soliton, rogue wave and breather solutions of a generalized Fokas–Lenells equation

Abstract

Under investigation is a generalized Fokas–Lenells equation, which is the first nontrivial equation in the negative hierarchy flow associated with a 2 × 2 matrix spectral problem involving two potentials. Based on the $N$-fold classical Darboux transformation, the ($n$, $N$-$n$)-fold generalized Darboux transformation is constructed by means of the limit technique. As an application, the higher-order localized wave solution in a compact determinant form with arbitrary order is derived for the generalized Fokas–Lenells equation. The bright-dark soliton solutions from first to third order under a zero background are obtained. Specifically, unlike the bright rogue wave and breather solutions of the standard Fokas–Lenells equation, the bright-dark and bright-bright rogue wave solutions as well as breather solutions from first to third order under a nonzero background of the generalized Fokas–Lenells equation are presented. Furthermore, the hybrid rogue wave-breather solutions from second to third order are given. The dynamical behaviors of these explicit solutions are all displayed graphically.