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 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.

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 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 Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

A weakly coupled Hirota equation and its rogue waves

2019 ◽  
Vol 34 (22) ◽  
pp. 1950179 ◽  
Author(s):  
Huijuan Zhou ◽  
Chuanzhong Li
Keyword(s):  
Optical Fibers ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Taylor Series Expansion ◽  
Rogue Waves ◽  
Nls Equation ◽  
Hirota Equation ◽  
Weakly Coupled ◽  
Parameter Values ◽  
Higher Order Dispersion

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text] and [Formula: see text] to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ([Formula: see text], [Formula: see text]) generated from the known solution ([Formula: see text], [Formula: see text]). The new solutions ([Formula: see text], [Formula: see text]) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

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 the higher-order semirational solutions for a non-linear Schrödinger system in a birefringent fiber

2021 ◽  
pp. 2150013
Author(s):  
Dan-Yu Yang ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Yu-Qiang Yuan ◽  
Chen-Rong Zhang ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Phase Modulation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Four Wave Mixing ◽  
Second Order ◽  
Wave Mixing ◽  
Third Order ◽  
Non Linear ◽  
Schrödinger System

Temporal birefringent effects in the fibers change the crosstalk behaviors inside and between the fiber cores in the linear and non-linear optical power areas. This paper studies a non-linear Schrödinger system with the four-wave mixing term, which describes the optical solitons in a birefringent fiber. We construct the generalized Darboux transformation, and acquire the higher-order semirational solutions consisting of the second- and third-order semirational solutions, which represent the complex amplitudes of the electric fields in the two orthogonal polarizations. We acquire the interactions between/among the two/three solitons. Such interactions are elastic and generate the rogue waves around the interacting regions. We obtain the interactions among the second-/third-order rogue waves and two/three solitons, respectively. When [Formula: see text] decreases, amplitude of the second-order rogue wave increases, with [Formula: see text] and [Formula: see text] accounting for the self-phase modulation and cross-phase modulation, respectively, while [Formula: see text] representing the four-wave mixing effect. With [Formula: see text] kept invariant, when [Formula: see text] increases and [Formula: see text], amplitudes of the second-order rogue wave and two bright solitons increase, while when [Formula: see text] increases and [Formula: see text], amplitudes of the second-order rogue wave and two dark solitons increase, with [Formula: see text] and [Formula: see text] being the constants.

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