Generalized Darboux transformation and nonlinear analysis of higher-order localized wave solutions

2021 ◽  
Author(s):  
N. Song ◽  
X. Y. Zhao ◽  
N. Shi
Keyword(s):  
Nonlinear Analysis ◽  
Darboux Transformation ◽  
Higher Order ◽  
Wave Solutions ◽  
Generalized Darboux Transformation

Generalized Darboux transformation and higher-order rogue wave solutions to the Manakov system

2021 ◽  
Author(s):  
Serge P. Mukam ◽  
Souleymanou Abbagari ◽  
Alphonse Houwe ◽  
Victor K. Kuetche ◽  
Mustafa Inc ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Higher Order ◽  
Lax Pairs ◽  
Wave Solutions ◽  
Free Parameters ◽  
Generalized Darboux Transformation ◽  
Rogue Wave Solutions ◽  
Physical Phenomena

In this paper, we propose a recursive Darboux transformation in a generalized form of a focusing vector Nonlinear Schrödinger Equation (NLSE) known as the Manakov System. We apply this generalized recursive Darboux transformation to the Lax-pairs of this system in view of generating the Nth-order vector generalization rogue wave solutions with a rule of iteration. We discuss from first- to three-order vector generalizations of rogue wave solutions while illustrating these features with some 3D, 2D graphical depictions. We illustrate a clear connection between higher-order rogue wave solutions and their free parameters for better understanding the physical phenomena described by the Manakov system

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

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.

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.

scholarly journals Construction of Higher-Order Smooth Positons and Breather Positons via Hirota's Bilinear Method

2021 ◽  
Author(s):  
Zhao Zhang ◽  
Biao Li ◽  
Junchao Chen ◽  
QI GUO
Keyword(s):  
Soliton Solution ◽  
Darboux Transformation ◽  
Mathematical Expression ◽  
Higher Order ◽  
Second Order ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Advantages And Disadvantages ◽  
De Vries ◽  
Generalized Darboux Transformation

Abstract Based on the Hirota's bilinear method, a more classic limit technique is perfected to obtain second-order smooth positons. Immediately afterwards, we propose an extremely ingenious limit approach in which higher-order smooth positons and breather positons can be quickly derived from N-soliton solution. Under this ingenious technique, the smooth positons and breather positons of the modified Korteweg-de Vries system are quickly and easily derived. Compared with the generalized Darboux transformation, the approach mentioned in this paper has the following advantages and disadvantages: the advantage is that it is simple and fast; the disadvantage is that this method cannot get a concise general mathematical expression of nth-order smooth positons.

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