N-fold Darboux transformation of the two-component Kundu–Eckhaus equations and non-symmetric doubly localized rogue waves

Abstract

The two-component Kundu–Eckhaus (KE) equations were introduced for the first time in 1999. Very recently, the two-component KE equations considered as a model of describing the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media have been intensively studied. In this paper, we construct an analytical and explicit representation of the Darboux transformation (DT) for the two-component KE equations. Compared with the DT constructed by researchers before, the DT here is expressed by the initial eigenfunctions, spectral parameters, and ‘seed’ solution. As applications of DT, the explicit expressions of non-symmetric rogue wave of two-component KE equations and KE equation are displayed, and the differences between the non-symmetric and symmetric rogue wave for the KE equation are discussed in detail.