scholarly journals Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

2019 ◽  
Vol 99 (5) ◽  
Author(s):  
A. Ankiewicz ◽  
Mahyar Bokaeeyan ◽  
N. Akhmediev
Keyword(s):  
Shallow Water ◽  
Vries Equation ◽  
Rogue Waves ◽  
De Vries ◽  
Complex Solutions

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 Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

2014 ◽  
Vol 90 (1) ◽  
Author(s):  
Anna Karczewska ◽  
Piotr Rozmej ◽  
Eryk Infeld
Keyword(s):  
Shallow Water ◽  
Vries Equation ◽  
Soliton Dynamics ◽  
De Vries

scholarly journals Rational Solutions of Multi-component Nonlinear Schrödinger Equation and Complex Modified KdV Equation

2021 ◽  
Author(s):  
Lihong Wang ◽  
Jingsong He ◽  
Róbert Erdélyi
Keyword(s):  
Explicit Formula ◽  
Darboux Transformation ◽  
Critical Condition ◽  
Vries Equation ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Modified Kdv Equation ◽  
De Vries

In this paper, the critical condition to achieve rational solutions of the multi-component nonlinear Schr\”odinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions,an explicit formula of the $n$th-order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the $n$th-order rogue waves is summarized. This conjecture also holds for rogue waves of the multi-component complex modified Korteweg-de Vries equation. Finally, the semi-rational solutions of the Manakov system are discussed.

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

Rogue waves on the periodic background in the higher-order modified Korteweg-de Vries equation

2020 ◽  
pp. 2150081
Author(s):  
Fa Chen ◽  
Hai-Qiang Zhang
Keyword(s):  
Vries Equation ◽  
Algebraic Method ◽  
Rogue Waves ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Higher Order ◽  
Jacobi Elliptic Function ◽  
Order Model ◽  
Periodic Wave Solutions ◽  
De Vries

In this paper, we investigate the higher-order modified Korteweg–de Vries (mKdV) equation by using an algebraic method. On the background of the Jacobi elliptic function, we obtain the admissible eigenvalues and the corresponding non-periodic eigenfunctions of the spectral problem in this higher-order model. Then, with the aid of the Darboux transformation (DT), we derive the rogue dn- and cn-periodic wave solutions. Finally, we analyze the non-linear dynamics of two kinds of rogue periodic waves.

scholarly journals Few-cycle optical rogue waves: Complex modified Korteweg–de Vries equation

2014 ◽  
Vol 89 (6) ◽  
Author(s):  
Jingsong He ◽  
Lihong Wang ◽  
Linjing Li ◽  
K. Porsezian ◽  
R. Erdélyi
Keyword(s):  
Vries Equation ◽  
Rogue Waves ◽  
De Vries
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