scholarly journals Novel Particular Solutions, Breathers, and Rogue Waves for an Integrable Nonlocal Derivative Nonlinear Schrödinger Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Author(s):  
Yali Shen ◽  
Ruoxia Yao
Keyword(s):  
Darboux Transformation ◽  
Taylor Expansion ◽  
Rogue Wave ◽  
Rogue Waves ◽  
First Order ◽  
Particular Solutions ◽  
Wave Solutions ◽  
Determinant Representation ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Dnls Equation

A determinant representation of the n -fold Darboux transformation for the integrable nonlocal derivative nonlinear Schödinger (DNLS) equation is presented. Using the proposed Darboux transformation, we construct some particular solutions from zero seed, which have not been reported so far for locally integrable systems. We also obtain explicit breathers from a nonzero seed with constant amplitude, deduce the corresponding extended Taylor expansion, and obtain several first-order rogue wave solutions. Our results reveal several interesting phenomena which differ from those emerging from the classical DNLS equation.

scholarly journals Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
N. Song ◽  
W. Zhang ◽  
P. Wang ◽  
Y. K. Xue
Keyword(s):  
Darboux Transformation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Wave Solutions ◽  
Generalized Darboux Transformation ◽  
Rogue Wave Solutions ◽  
Fifth Order

The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation. The conclusions will be a supportive tool to study the rogue waves better.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

scholarly journals General rogue wave solutions of the coupled Fokas–Lenells equations and non-recursive Darboux transformation

2019 ◽  
Vol 475 (2224) ◽  
pp. 20180806 ◽  
Author(s):  
Yanlin Ye ◽  
Yi Zhou ◽  
Shihua Chen ◽  
Fabio Baronio ◽  
Philippe Grelu
Keyword(s):  
Darboux Transformation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Background Field ◽  
Double Root ◽  
Transformation Technique ◽  
Wave Dynamics ◽  
Wave Solutions ◽  
Future Experimental Study ◽  
Rogue Wave Solutions

We formulate a non-recursive Darboux transformation technique to obtain the general n th-order rational rogue wave solutions to the coupled Fokas–Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.

scholarly journals The Darboux transformation of the Kundu–Eckhaus equation

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150236 ◽  
Author(s):  
Deqin Qiu ◽  
Jingsong He ◽  
Yongshuai Zhang ◽  
K. Porsezian
Keyword(s):  
Frequency Shift ◽  
Darboux Transformation ◽  
Taylor Expansion ◽  
Rogue Wave ◽  
Explicit Representation ◽  
The Self ◽  
First Order ◽  
Analytical Formulae ◽  
Degenerate Eigenvalues ◽  
The Impact

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution and n -fold DT T n are given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues λ 2 k − 1 → λ 1 = − 1 2 a + β c 2 + i c , k =1,2,3,…, all these parameters will be defined latter. These solutions have a parameter β , which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the β on the RW solution. We observe two interesting results on localization characters of β , such that if β is increasing from a /2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if β is increasing to a /2. We define an area of the RW solution and find that this area associated with c =1 is invariant when a and β are changing.

Generalized Darboux transformation and rogue wave solution of the coherently-coupled nonlinear Schrödinger system

2016 ◽  
Vol 30 (13) ◽  
pp. 1650208 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Sha-Sha Yuan ◽  
Yue Wang
Keyword(s):  
Darboux Transformation ◽  
Wave Solution ◽  
Rogue Wave ◽  
Rogue Waves ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Rogue Wave Solution ◽  
Bright Soliton Solution ◽  
Generalized Darboux Transformation ◽  
Schrödinger System

In this paper, the generalized Darboux transformation for the coherently-coupled nonlinear Schrödinger (CCNLS) system is constructed in terms of determinant representations. Based on the Nth-iterated formula, the vector bright soliton solution and vector rogue wave solution are systematically derived under the nonvanishing background. The general first-order vector rogue wave solution can admit many different fundamental patterns including eye-shaped and four-petaled rogue waves. It is believed that there are many more abundant patterns for high order vector rogue waves in CCNLS system.

scholarly journals Multi-Elliptic Rogue Wave Clusters of the Nonlinear Schrodinger Equation on Different Backgrounds

2021 ◽  
Author(s):  
Stanko N Nikolic ◽  
Sarah Al Washahi ◽  
Omar A. Ashour ◽  
Siu A. Chin ◽  
Najdan B. Aleksic ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Intensity Distribution ◽  
High Intensity ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Radial Symmetry ◽  
Hirota Equation ◽  
Narrow Peak ◽  
Infinite Hierarchy ◽  
First Order

Abstract In this work we analyze the multi-elliptic rogue wave clusters as new solutions of the nonlinear Schr\"odinger equation (NLSE). Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme of order $n$ with the first $m$ evolution shifts that are equal, nonzero, and eigenvalue-dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of $n-2m$ order appears at the origin of the $(x,t)$ plane and can be considered as the central rogue wave of the cluster. We show that the high-intensity narrow peak, with characteristic intensity distribution in its vicinity, is enclosed by $m$ ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a periodic background, we utilize the modified Darboux transformation scheme to build these solutions on a Jacobi elliptic dnoidal background. We analyze the minor semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the $(x,t)$ plane is violated when the shift values are increased above a threshold. We apply the same analysis on Hirota equation, to examine the influence of a free real parameter and Hirota operator on the cluster appearance. The same analysis can be extended to the infinite hierarchy of extended NLSEs.

scholarly journals Rogue Wave and Multiple Lump Solutions of the (2+1)-Dimensional Benjamin-Ono Equation in Fluid Mechanics

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Zhonglong Zhao ◽  
Lingchao He ◽  
Yubin Gao
Keyword(s):  
Rogue Wave ◽  
Rogue Waves ◽  
Wave Form ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
First Order ◽  
Wave Solutions ◽  
Rogue Wave Solutions ◽  
Lump Wave

In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8-lump solutions are presented, respectively. The 3-lump wave has a “triangular” structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.

Multi-Soliton and Rogue-Wave Solutions of the Higher-Order Hirota System for an Erbium-Doped Nonlinear Fiber

2014 ◽  
Vol 69 (10-11) ◽  
pp. 521-531 ◽  
Author(s):  
Da-Wei Zuo ◽  
Yi-Tian Gao ◽  
Yu-Hao Sun ◽  
Yu-Jie Feng ◽  
Long Xue
Keyword(s):  
Wave Propagation ◽  
Darboux Transformation ◽  
Rogue Wave ◽  
Higher Order ◽  
Soliton Interaction ◽  
Nls Equation ◽  
First Order ◽  
Wave Solutions ◽  
Erbium Doped ◽  
Nonlinear Fiber

AbstractThe nonlinear Schrödinger (NLS) equation appears in fluid mechanics, plasma physics, etc., while the Hirota equation, a higher-order NLS equation, has been introduced. In this paper, a higher-order Hirota system is investigated, which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order dispersion. By virtue of the Darboux transformation and generalized Darboux transformation, multi-soliton solutions and higher-order rogue-wave solutions are derived, beyond the published first-order consideration.Wave propagation and interaction are analyzed: (i) Bell-shape solitons, bright- and dark-rogue waves are found; (ii) the two-soliton interaction is elastic, i. e., the amplitude and velocity of each soliton remain unchanged after the interaction; (iii) the coefficient in the system affects the direction of the soliton propagation, patterns of the soliton interaction, distance, and direction of the first-order rogue-wave propagation, as well as the range and direction of the second-order rogue-wave interaction.

Breathers and multiple rogue waves solutions of the (3+1)-dimensional Jimbo–Miwa equation

2021 ◽  
pp. 2150183
Author(s):  
Hong-Yi Zhang ◽  
Yu-Feng Zhang
Keyword(s):  
Dynamic Characteristics ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Wave Solutions ◽  
Hirota Bilinear

In this paper, we construct the breathers of the (3+1)-dimensional Jimbo–Miwa (JM) equation by means of the Hirota bilinear method, then based on the Hirota bilinear method with a new ansatz form, the multiple rogue wave solutions are constructed. Here, we discuss the general breathers, first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then we draw the 3- and 2-dimensional plots to illustrate the dynamic characteristics of breathers and multiple rogue waves. These interesting results will help us better reveal (3+1)-dimensional JM equation evolution mechanism.

Breathers and Rogue Waves for the Fourth-Order Nonlinear Schrödinger Equation

2017 ◽  
Vol 72 (4) ◽  
pp. 339-344
Author(s):  
Yan Zhang ◽  
Yinping Liu ◽  
Xiaoyan Tang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Darboux Transformation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

Abstract:In this article, a generalized Darboux transformation for the fourth-order nonlinear Schrödinger equation is constructed in terms of Darboux matrix method. Subsequently, breathers and the Nth-order rogue wave solutions of this equation are explicitly given in the light of the obtained Darboux transformation. Finally, we concretely discuss the dynamics of the obtained rogue waves, which are also demonstrated by some figures.

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