^{1,a)}, Shuwei Xu

^{2}and Yi Cheng

^{2}

I have read the terms and conditions of "Ratings and Commenting".

### Abstract

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 Tn
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.

This work is supported by the NSF of China under Grant No.10971109 and 11271210. Jingsong He is also supported by the Natural Science Foundation of Ningbo under Grant No.2011A610179, and K.C. Wong Magna Fund in Ningbo University. J. He thanks sincerely Prof. A.S. Fokas for arranging the visit to Cambridge University in 2012-2013 and for many useful discussions.

I. INTRODUCTION II. DARBOUX TRANSFORMATION A. One-fold Darboux transformation of the WKI system B. n-fold Darboux transformation for WKI system C. Reduction of the Darboux transformation for WKI system III. SMOOTHNESS OF THE SOLUTIONS

*q*

^{[2k]}A. Non-degenerate case B. Double degeneration case IV. RATIONAL SOLUTIONS GENERATED BY 2K-FOLD DEGENERATE DARBOUX TRANSFORMATION A. Asymptotic behavior of rational 1-order solution B. Analytical forms and localization of the higher order rogue wave solutions V. CONCLUSIONS AND DISCUSSIONS

##### G02F1/35

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### Abstract

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 Tn
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.

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