Symmetry reductions of the ( 3 + 1 ) $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

Liu, Yamin; Teng, Qingyong; Tai, Weipeng; Zhou, Jianping; Wang, Zhen

doi:10.1186/s13662-019-2017-4

Cited by 8 publications

(3 citation statements)

References 40 publications

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“…Diverse approaches have been attained to study such models aiming to construct their wave solutions and discuss their behavior and properties. The most popular approaches are: the improved auxiliary differential equation method [4], the extended direct algebraic method [5,6], Lie point symmetries, reductions, conservation laws [7][8][9], the solitary wave ansatze [10], the reductive perturbation method [11][12][13], the homogeneous balance method [14], the direct symmetry method [15], the extended ( ) ¢ -G G 2 expansion method [16,17], the sine-Gordon expansion method and the ( ) ¢ -G 1…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

Elbrolosy

2024

Phys. Scr.

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In this paper, the extended (3 + 1)-dimensional Zakharov-Kuznetsov equation, which describes the propagation of ion-acoustic waves in a magnetic environment, is investigated. Due to the exposure of the propagation to unpredictable factors, the stochastic model is assessed including the Brownian process, in addition to including the recent concept of truncated M-fractional derivative. A fractional stochastic transformation is applied to transform the model into an integer-order ordinary differential equation which in turn is equivalent to a conservative Hamiltonian model. Novel solutions, such as hyperbolic, trigonometric, and Jacobian elliptic functions, are established by employing both of the qualitative analysis of dynamical systems and the first integral of the Hamiltonian model. We explore and graphically display the effects of the fractional derivative order and noise intensity on the solutions structures. In the deterministic instance, i.e., in the absence of noise, solitary and cnoidal solutions among other traveling wave solutions of the Zakharov-Kuznetsov equation, are derived. Further, it is found that the curvature of the wave disturbs and the surface turns substantially flat by increasing the value of noise. While the curve in all cases loses its characteristic shape and degenerates into another deterministic shape by changing the fractional derivative order.

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Section: Introductionmentioning

confidence: 99%

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

Elbrolosy

2024

Phys. Scr.

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“…The mZK equation have solutions that symbolize plane-periodic and solitary traveling waves propagating. It is noted that the mZK equation is a high dimensional nonlinear evolution equation and, thus, the study of its reduction problem is of theoretical interest [47]. Park et al [48] applied modified Khater method to equation.…”

Section: Introductionmentioning

confidence: 99%

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Saha

Karakoç

Khalid

2023

Mathematical Sciences and Applications E-Notes

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The aim of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher order dispersion term. For this purpose, first and second simplest methods are used to build soliton solutions of travelling wave solutions. Furthermore, bifurcation behavior of traveling waves including new type of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and benefical tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher order dispersion will add some value in the literature of mathematical and plasma physics.

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“…The Lie symmetry technique is one of the most useful techniques to conclude to solutions of nonlinear FPDEs, generally, Lie symmetries might be used to reduce the order of the original equation (system of equations) as well as the number of independent variables [5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Lie symmetry reductions and conservation laws for fractional order coupled KdV system

2020

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Lie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system.In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.

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Symmetry reductions of the ( 3 + 1 ) $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

Cited by 8 publications

References 40 publications

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Lie symmetry reductions and conservation laws for fractional order coupled KdV system

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