Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation

Under investigation in this paper is the Kundu equation, which may be used to describe the propagation process of ultrashort optical pulses in nonlinear optics. The modulational instability of the plane-wave for the possible reason of the formation of the rogue wave (RW) is studied for the system. Based on our proposed generalized perturbation (n,N − n)-fold Darboux transformation (DT), some new higher-order implicit RW solutions in terms of determinants are obtained by means of the generalized perturbation (1,N − 1)-fold DT, when choosing different special parameters, these results will reduce to the RW solutions of the Kaup–Newell (KN) equation, Chen–Lee–Liu (CLL) equation and Gerjikov–Ivanov (GI) equation, respectively. The relevant wave structures are shown graphically, which display abundant interesting wave structures. The dynamical behaviors and propagation stability of the first-order and second-order RW solutions are discussed by using numerical simulations, the higher-order nonlinear terms for the Kundu equation have an impact on the propagation instability of the RW. The method can also be extended to find the higher-order RW or rational solutions of other integrable nonlinear equations.

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