Rogue waves on the general periodic traveling wave background for an extended modified Korteweg–de Vries equation

2021 ◽  
Author(s):  
Yu Lou ◽  
Yi Zhang ◽  
Rusuo Ye
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Rogue Waves ◽  
De Vries ◽  
Periodic Traveling Wave

scholarly journals Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation

2000 ◽  
Vol 24 (6) ◽  
pp. 371-377 ◽  
Author(s):  
Kenneth L. Jones ◽  
Xiaogui He ◽  
Yunkai Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equations ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
De Vries ◽  
Periodic Traveling Wave

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of(uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation,(ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

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.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation

2016 ◽  
Vol 26 (8) ◽  
pp. 084312 ◽  
Author(s):  
Xiao-Jun Yang ◽  
J. A. Tenreiro Machado ◽  
Dumitru Baleanu ◽  
Carlo Cattani
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
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.

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