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The mixed nonlinear Schrödinger (MNLS) equation is a model for the propagation of the Alfvén wave in plasmas and the ultrashort light pulse in optical fibers with two nonlinear effects of self-steepening and self phase-modulation(SPM), which is also the first non-trivial flow of the integrable Wadati-Konno-Ichikawa(WKI) system. The determinant representation of a n-fold Darboux transformation(DT) for the MNLS equation is presented. The smoothness of the solution [2] generated by is proved for the two cases (non-degeneration and double-degeneration ) through the iteration and determinant representation. Starting from a periodic seed(plane wave), rational solutions with two parameters and of the MNLS equation are constructed by the DT and the Taylor expansion. Two parameters denote the contributions of two nonlinear effects in solutions. We show an unusual result: for a given value of , the increasing value of can damage gradually the localization of the rational solution, by analytical forms and figures. A novel two-peak rational solution with variable height and a non-vanishing boundary is also obtained.


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