Higher-order rogue wave solutions of the Kundu–Eckhaus equation

Abstract

In this paper, we investigate higher-order rogue wave solutions of the Kundu-Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu-Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu-Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu-Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.

Rogue wave spectra of the Kundu-Eckhaus equation

“…This solution and some other analytical solutions are given in [8][9][10]. This first order rational rogue wave is basically a skewed Peregrine soliton of the NLSE.…”

“…where ψ is complex amplitude, x, t are the spatial and temporal variables and i is the imaginary number [8].…”

“…For the cubic NLSE, the first and higher order rational rogue wave solutions can be seen in [12]. Second and the higher order rational solutions of the KEE and a hierarchy of obtaining those rational solutions based on Darboux transformations are presented in [8]. They are basically skewed rogue waves obtained by the gauge transforming the rogue wave solutions of the cubic NLSE.…”

Rogue waves and W-shaped solitons in the multiple self-induced transparency system

“…2(b). The analogous phenomena produced by the higher-order effects have been comprehensively studied for rogue waves in the Hirota equation [24], Sasa-Satsuma equation [25] and Kundu-Eckhaus equation [36,39], while for the coupled system without any higher-order terms they have been rarely reported.…”

“…The purpose of this paper is mainly concentrated on two aspects: (1) we construct an Nth-order rational solution in a compact determinant representation by taking advantage of the generalized DT approach [33][34][35][36]; (2) With the aid of the analytical rational solution and modulation instability (MI), dynamics of the optical rogue waves, as well as the stationary and nonstationary W-shaped solitons from first to third order are presented. In particular, it is importantly found that, the nonstationary W-shaped solitons can only exist in the multiple SIT system, while for the single system they are impossible to appear.…”