Periodic travelling waves and rogue waves for the higher-order modified Korteweg-de Vries equation

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.

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Rogue waves on the general periodic travelling waves background for an extended modified Korteweg-de Vries equation

10.22541/au.161449581.12602619/v1 ◽

2021 ◽

Author(s):

Yi Zhang ◽

Yu Lou ◽

RS Ye

Keyword(s):

Vries Equation ◽

Travelling Waves ◽

Three Dimensional ◽

Rogue Waves ◽

Travelling Wave ◽

Periodic Waves ◽

Travelling Wave Solutions ◽

Third Order ◽

Periodic Travelling Wave ◽

De Vries

Under consideration in this paper is rogue waves on the general periodic travelling waves background of an integrable extended modified Korteweg-de Vries equation. The general periodic travelling wave solutions are presented in terms of the sub-equation method. By means of the Darboux transformation and the nonlinearization of the Lax pair, we present the first-, second- and third-order rogue waves on the general periodic travelling waves background. Furthermore, the dynamic behaviors of rogue periodic waves are elucidated from the viewpoint of three-dimensional structures.

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

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0094 ◽

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.

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Gauge transformation and the higher order Korteweg–de Vries equation

Journal of Mathematical Physics ◽

10.1063/1.528068 ◽

1988 ◽

Vol 29 (2) ◽

pp. 308-314 ◽

Cited By ~ 7

Author(s):

Yu‐kun Zheng ◽

W. L. Chan

Keyword(s):

Gauge Transformation ◽

Vries Equation ◽

Higher Order ◽

De Vries

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Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

Modern Physics Letters A ◽

10.1142/s0217732319500548 ◽

2019 ◽

Vol 34 (07n08) ◽

pp. 1950054

Author(s):

H. Wajahat A. Riaz

Keyword(s):

Evolution Equations ◽

Nonlinear Evolution ◽

Vries Equation ◽

Soliton Solutions ◽

Higher Order ◽

Nonlinear Evolution Equations ◽

Order Effects ◽

Darboux Transformations ◽

De Vries ◽

Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

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