  The mixed nonlinear Schr\"odinger (MNLS) equation is a model for the
propagation of the Alfv\'en 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.
