On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

Akbulut, Arzu; Almusawa, Hassan; Kaplan, Melike; Osman, M. S.

doi:10.3390/sym13050765

Cited by 27 publications

(5 citation statements)

References 41 publications

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“…Substituting the equation (23) with equation ( 21) into the equation (3) and collecting the coefficient of R(ξ) j = 0 ( j = 0,K,3), we obtain the following algebraic system;…”

Section: Implementation Of Nkmmentioning

confidence: 99%

“…For this reason, the importance of the relevant model becomes high. Some studies on the equation are as follows: Exact solutions of time-fractional mKdV-ZK with undetermined coefficients method [16], Painleve analysis and wave profiles of model [17], obtaining soliton solution via modified extended mapping method [18], via ( ) ( ) tan 2 f x method [19], improved fractional sub-equation method [20], first integral method [21], functional variable method [22], investigation of the conservation laws on the model [23] and Lie symmetry analysis of the model [24] etc.…”

Section: Introductionmentioning

confidence: 99%

“…Substituting the SET 2,1 into the equation(23) with equation(22) and wave transformation in equation (2), we obtain the following solutions of equation (1):…”

mentioning

confidence: 99%

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Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Onder,

Secer,

Bayram

2023

Phys. Scr.

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In this paper, we explore analytical solutions for the (3+1)-dimensional time-fractional modified Korteweg–de Vries Zakharov-Kuznetsov equation, which incorporates a conformable derivative. Our interest in this model is driven by its significant role in simulating ion-acoustic waves in magnetized plasma. We adopt the unified Riccati equation expansion method and the new Kudrashov method to discover soliton solutions. Our approach uncovers various soliton types, such as kink, singular, periodic-singular, and bright solitons. We conduct a thorough analysis of how different parameters affect wave propagation, enhancing our study with descriptive figures and insightful observations. Furthermore, we delve into the modulation instability characteristic of this model. The influence of specific parameters, like wave number and the order of the conformable derivative, on wave dynamics is demonstrated through detailed visualizations. We also present 2D and 3D graphical representations of these solutions.

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“…Substituting the equation (23) with equation ( 21) into the equation (3) and collecting the coefficient of R(ξ) j = 0 ( j = 0,K,3), we obtain the following algebraic system;…”

Section: Implementation Of Nkmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Onder,

Secer,

Bayram

2023

Phys. Scr.

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“…The study of exact solutions of such mathematical models is the most exciting area of research investigation. To obtain the exact solutions of such mathematical models, recently, researchers follow different analytical methods, such as tangent hyperbolic method [2], modified extended direct algebraic method [3,4], ðG′/G 2 Þ -expansion method [5], Kudryashov method [6], Lie symmetry method [7], extended trial equation method [8], singular manifold method [9], sinh-Gordon expansion method [10], ðG′/GÞ-expansion method [11], generalized ð G′/GÞ-expansion method [12], generalized-improved ðG′/ GÞ-expansion method [13], extended generalized ðG ′ /GÞ -expansion method [14,15], and ðG′/G, 1/GÞ expansion method [16].…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions of the Generalized ZK and Gardner Equations by Extended Generalized $(G^{'} / G)$ -Expansion Method

Mohanty

Dev

Sahoo³

et al. 2023

Advances in Mathematical Physics

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In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalized G ′ / G expansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalized G ′ / G expansion technique. The extended generalized G ′ / G expansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial of G ′ / G . The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.

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“…The exact solutions of NLPDEs symbolically and physically reveal the mechanisms of many nonlinear complex phenomena. Several techniques have been developed for investigating exact wave structures of NLPDEs in the shape of soliton, traveling wave, and elliptic function solutions 1–30 . Muto et al have firstly submitted the nonlinear mathematical model of the interaction of deoxyribonucleic acid (DNA) with an external microwave field which was proposed as

ϕ_{t t} = C^{2} ϕ_{z z} - (ε / C^{2}) ϕ_{z z t t} + δ {(ϕ_{z}^{2})}_{z},

where ϕ ( z , t ) describes longitudinal displacements in DNA 31,32 .…”

Section: Introductionmentioning

confidence: 99%

Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

Bilal

Younas

Ren

2021

Math Methods in App Sciences

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In this research, we study analytically the double‐chain model. The model consists of two long elastic homogeneous strands (or rods), which represent two polynucleotide chains of the deoxyribonucleic acid molecule, connected with each other by an elastic membrane (or some linear springs) representing the hydrogen bonds between the base pairs of the two chains. The new extended direct algebraic method and the generalized Kudryashov method are successfully utilized to discuss the exact soliton solutions to the double‐chain model of deoxyribonucleic acid that plays an important role in biology. The solutions obtained by these mechanisms can be divided into solitary, singular, kink, single wave, combine behavior as well as hyperbolic, plane wave, and trigonometric solutions with arbitrary parameters. Some solutions have been exemplified by graphics to understand the physical meaning of the DNA model. The accomplished solutions seem with all essential constraint conditions, which are obligatory for them to subsist. Hence, our techniques via fortification of symbolic computations provide an active and potent mathematical implement for solving diverse benevolent nonlinear wave problems. The results show that the system theoretically has extremely rich exact wave structures of biological relevance.

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On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

Cited by 27 publications

References 41 publications

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Exact Solutions of the Generalized ZK and Gardner Equations by Extended Generalized $(G^{'} / G)$ -Expansion Method

Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

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On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

Cited by 27 publications

References 41 publications

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

Exact Solutions of the Generalized ZK and Gardner Equations by Extended GeneralizedG′/G-Expansion Method

Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

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Exact Solutions of the Generalized ZK and Gardner Equations by Extended Generalized $(G^{'} / G)$ -Expansion Method