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 T2k 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.
Highlights
Rogue wave (RW) has been introduced and become an interesting objective in the investigation of oceanography [1, 2](and references therein), starting with modeling a short-lived large amplitude wave in ocean
One of widely accepted prototypes of rogue wave in one dimensional space and time is considered as Peregrine soliton [12,13,14] of the nonlinear Schrodinger equation (NLS), which usually takes the form of a single dominant peak accompanied by one deep cave at each side in a plane with a nonzero boundary
Considering many wave propagation phenomena described by integrable equations in some ideal conditions, the effects of small perturbations on the mixed NLS (MNLS) equation are study by the direct soliton perturbation theory [53] and the perturbation theory based on the inverse scattering transform [54, 55]
Summary
Rogue wave (RW) has been introduced and become an interesting objective in the investigation of oceanography [1, 2](and references therein), starting with modeling a short-lived large amplitude wave in ocean. The characteristic property of the RW is localization in both space and time directions in a nonzero plane The existence of this solution is due to modulation instability of the NLS equation [12,15,16,17,18]. Considering many wave propagation phenomena described by integrable equations in some ideal conditions, the effects of small perturbations on the MNLS equation are study by the direct soliton perturbation theory [53] and the perturbation theory based on the inverse scattering transform [54, 55].
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