DynamicsofcontrollableroguewavesinageneralizednonautonomousKundu-Eckhausequation资源-CSDN文库

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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

广义非自治Kundu-Eckhaus方程中可操控怪波

的动力学研究

马笑霄

，赵立臣

，刘冲

，杨战营

，杨文力

西北大学物理学系，西安　710069

西北大学近代物理研究所，西安　710069

摘要：本文解析地研究了广义非自治Kundu-Eckhaus方程中的怪波。其中，通过给出怪波的波

峰、波谷、轨迹和宽度的精确表达式，对怪波的动力学进行了细致的分析和研究。基于此，本

文详细地展示了一些具有鲜明特色怪波的演化特性，同时，怪波的动力学能够在不同的非线性

管理下很好地操控。通过本文的研究我们发现，当怪波的宽度值演化为最小值时，其波峰值恰

好达到最大值，对应的波谷值达到最小值。此外，由于所研究模型的可积性的限制，方程中的

高阶效应只对怪波的轨迹有影响。

关键词：怪波；动力学特性；非自治Kundu-Eckhaus方程；非线性管理

中图分类号： N93

Dynamics of controllable rogue waves in a

generalized nonautonomous Kundu-Eckhaus

equation

Ma Xiaoxiao

，Zhao Lichen

，Liu Chong

，Yang Zhanying

，Yang Wenli

Department of Physics, Northwest University, Xi’an 710069, China

Institute of Modern Physics, Northwest University, Xi’an 710069, China

Abstract: We analytically study rogue waves in a generalized nonautonomous

Kundu-Eckhaus equation. Dynamics of the rogue waves are investigated through the precise

expressions of their peak, valley, trajectory, and width. Based on this, the properties of a few

speciﬁc rogue waves are demonstrated in detail, and the dynamical evolution of rogue waves

can be well controlled under diﬀerent nonlinearity management. It shows that the p eak

reaches its maximum and the valley becomes minimized when the width evolves to the

minimum value. Moreover, we ﬁnd that the higher-order eﬀects here achieve balance due to

the integrability, and they only inﬂuence the rogue waves’ trajectory.

Key words: Rogue waves; dynamics property; nonautonomous Kundu-Eckhaus equation;

nonlinearity management

基金项目： The National Natural Science Foundation of China (NSFC)(Grant No. 11347605), and the ministry

of education doctoral program funds (Grant No. 20126101110004)

作者简介： Ma Xiaoxiao（1990-），female，master，major research direction：nonlinear physics. Correspondence

author：Yang Zhanying（1973-），male，professor，major research direction：nonlinear physics, zyyang@nwu.edu.cn.

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

Rogue waves (RWs) are huge single waves in oceans, and their appearance are quite unex-

pected and mysterious in origin [1, 2, 3, 4, 5]. This destructive phenomenon has been identiﬁed

in nonlinear optics [6], plasmas [7], superﬂuids [8], etc. [9, 10, 11, 12]. Recent studies have

shown that nonlinear theory is suitable to RWs description [13]. More speciﬁcally, a rational

solution of the standard nonlinear Schr¨odinger equation (NLSE) provides a formal prototypical

description of RWs [14]. This solution is localized in both space and time, and its peak gets

ampliﬁed through a maximal factor of 3 over a plane wave background. The characteristic of

this solution has been demonstrated experimentally in nonlinear ﬁber [15], water wave tank

[16], and plasma system [17].

In most cases, the dynamics of these nonlinear systems is more complicated than the

one described by the standard NLSE. Therefore, the higher-order eﬀects [18, 19] have to be

considered. For example, in ultrafast ﬁber telecommunication systems, higher-order eﬀects such

as higher-order disp ersion, self-steepening, self-frequency, and non-Kerr nonlinearity come into

play, and then the physical characteristics of the localized waves can change. Moreover, for real

experiments, there are several varying system parameters, including dispersion, nonlinearity,

and dissipation or gain, etc. [20], which opens a venue into experimental manipulation of RWs.

Therefore, the management and control play a key role in RWs studies since the pioneering

work of Serkin et al. [20], and Yan [21]. As Akhmediev et al. [22] point out, it is still an open

issue to study RWs in these more complicated cases. The existence of RWs has been recently

analyzed in some works [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], which concentrate mainly

on the construction of exact RWs solutions for some integrable higher-order NLSE models,

such as the Hirota equation [23, 24, 25, 26, 27], the Sasa-Satsuma equation [28, 29], and others

[30, 31, 32]. However, to our knowledge, the speciﬁc dynamics analysis and the description

of physical properties of the RWs, which have great signiﬁcance for the understanding of the

RWs’ physical mechanism, are rarely studied.

In this paper we consider a generalized nonautonomous Kundu-Eckhaus (KE) equation

which is a further modiﬁcation of the NLSE with two additional terms that are responsible for

the quintic nonlinearity and the cubic nonlinear dispersion. The exact RW solution is obtained

by the similarity transformation and Darboux transformation. In contrast to the analytical

method used in [26, 27, 31], we investigated the dynamical properties of the RWs analytically

through the precise expressions of their peak, valley, trajectory, and width. Especially, we

analyze the evolution properties of these precise expressions, and ﬁnd that the RWs can be

well controlled under diﬀerent nonlinearity management. Moreover, it is reported that the

higher-order eﬀects here only inﬂuence the RWs’ trajectories.

- 3 -

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