On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation

To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger (NLS) equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions.

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The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2018-0365 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wei Tan ◽

Zhao-Yang Yin

Keyword(s):

Differential Equations ◽

Wave Equations ◽

Rogue Wave ◽

Nonlinear Partial Differential Equations ◽

Rogue Waves ◽

Homoclinic Solutions ◽

Nonlinear Wave Equations ◽

Bilinear Method ◽

Wave Solutions ◽

Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

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Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

Nonlinear Dynamics ◽

10.1007/s11071-021-06578-x ◽

2021 ◽

Author(s):

Jun Yang ◽

Yan-Li Zhang ◽

Li-Yuan Ma

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Rogue Wave ◽

Higher Order ◽

Nonlinear Schrodinger Equation ◽

Discrete Nonlinear Schrödinger Equation ◽

Nonlinear Schrödinger ◽

Wave Solutions ◽

Rogue Wave Solutions

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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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