Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation

He, Bin; Meng, Qing

doi:10.1016/j.cnsns.2009.07.019

Cited by 40 publications

(11 citation statements)

References 17 publications

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“…Not only the existence of these solutions is proved, but also their concrete expressions are presented. Our work extends some previous results [10][11][12][13][14][15][16].…”

Section: Introductionsupporting

confidence: 90%

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The Bifurcations of Traveling Wave Solutions of the Kundu Equation

Liu

2013

Journal of Applied Mathematics

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“…Not only the existence of these solutions is proved, but also their concrete expressions are presented. Our work extends some previous results [10][11][12][13][14][15][16].…”

Section: Introductionsupporting

confidence: 90%

“…(2) is stable in [15]. He and Meng [16] investigated the exact traveling waves for (4) by using the bifurcation theory and the method of phase portraits analysis.…”

Section: Introductionmentioning

confidence: 99%

The Bifurcations of Traveling Wave Solutions of the Kundu Equation

Liu

2013

Journal of Applied Mathematics

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“…Lemma 1 (see [15]). Let ΨðξÞ be a continuous solution of system ( 9) for ξ ∈ ð−∞, + ∞Þ and lim ξ⟶∞ ΨðξÞ = m, lim ξ⟶−∞ ΨðξÞ = n:…”

Section: Bifurcations Of Phase Portraits Of System (9)mentioning

confidence: 99%

Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

Zhao

Han

2021

Advances in Mathematical Physics

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In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.

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“…us, by the bifurcation theory of planar dynamical system (see [17][18][19][20][21]), we can easily construct the exact solution of equation ( 1) depending on the parameters A and B.…”

Section: Phase Portraits Of System (1)mentioning

confidence: 99%

Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

Han

2021

Discrete Dynamics in Nature and Society

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In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

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Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation

Cited by 40 publications

References 17 publications

The Bifurcations of Traveling Wave Solutions of the Kundu Equation

The Bifurcations of Traveling Wave Solutions of the Kundu Equation

Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

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