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The rational solutions of the mixed nonlinear Schrödinger equation

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 $T_n$ of a n-fold Darboux transformation(DT) for the MNLS equation is presented. The smoothness of the solution $q^{[2k]}$ generated by $T_{2k}$ 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 $a$ and $b$ 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 $a$, the increasing value of $b$ 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.