Combined sinh-cosh-Gordon equation: Symmetry reductions, exact solutions and conservation laws

Khalique, Chaudry Masood; Magalakwe, Gabriel

doi:10.2989/16073606.2013.779969

Cited by 18 publications

(8 citation statements)

References 24 publications

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“…Similarly, the solutions of fractional differential equations have attracted the attention of researchers in many fields. Thus, many methods have been proposed one after another, such as the first integral method [34,35], (G'/G)expansion method [36,37], the Hirota method [38,39], the trial function method [40], the subequation method [41], and others [42][43][44][45][46]. In this paper, using extended Kudryashov method [47], the exact solution of (3+1)-dimensional TSF-ZK equation is obtained.…”

Section: Introductionmentioning

confidence: 99%

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Yang

2019

Complexity

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Dust plasma is a new field of physics which has developed rapidly in recent decades. The study of dust plasma has received much attention due to its importance in the environment of space and the Earth. Dust acoustic waves are generated because of the inertia of dust mass while the restoring force is provided by the thermal pressure of electrons and ions. Since dust acoustic waves were first reported theoretically in unmagnetized dust plasma by Rao et al., they have become a research hot spot. In this paper, the excitation of dust acoustic waves by a gravity field in a dust plasma is analyzed. According to the control equations of dust plasma motion and employing multiscale analysis and perturbation method, we have obtained a (3+1)-dimensional ZK model. Because of the space property of dust plasma, (3+1)-dimensional ZK equation is more suitable than KdV equation and (2+1)-dimensional ZK equation to describe the real dust acoustic waves. Then, the (3+1)-dimensional time-space fractional ZK (TSF-ZK) equation describing the fractal process of nonlinear dust acoustic waves is given for the first time. To further explore how dust acoustic waves change energy as they travel, we discuss the conservation laws of the new model. Moreover, we study the exact solution of (3+1)-dimensional TSF-ZK equation by using extended Kudryashov method. Finally, based on the exact solution, we further investigate the effect of the parameter k, the charge properties of dust particle Zd0, the fractional order values α, β, γ, and θ, the temperature Td, the gravity g, and the collision frequency β0 and β1 on the properties of dust acoustic waves by a gravity field in dust plasma.

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Section: Introductionmentioning

confidence: 99%

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Yang

2019

Complexity

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“…Nonlinear partial differential equations [8][9][10] play an important role in the field of Rossby waves. Many models have been derived [11][12][13][14], and many methods have been used to solve the nonlinear partial differential equations, such as the algebro-geometric method [15,16], Hirota method [17], Painlevé analysis method [18], Darboux transformations [19][20][21], Lie symmetry method [22][23][24] and so on [25][26][27][28][29]. Based on the classic Lie group method, the (2+1)-dimensional nonlinear inviscid barotropic nondivergent vorticity equation was studied by Huang and Lou [30], and the (3 + 1)-dimensional nonlinear Charney-Obukhov equation was studied by Kudryavtsev and Myagkov [31].…”

Section: Introductionmentioning

confidence: 99%

Symmetry analysis for three-dimensional dissipation Rossby waves

et al. 2018

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Rossby waves, belonging to the most important waves in the atmosphere and ocean, can affect the energy transfer of the atmosphere and ocean and have significant theoretical meaning and research value. In previous research performed with the theory and calculation method limit, the dissipation effect was commonly ignored. However, under the conditions of the weak linear approximation, the magnitude difference between nonlinear and dissipation is small, and the dissipation effect must be considered. In this paper, based on the classic Lie group approach, the (3 + 1)-dimensional quasi-geodetiophic vorticity equation with dissipation effect is solved. With the help of the solutions, we can better comprehend the influence of the dissipation effect on the propagation of Rossby waves.

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“…Gomez and Salas [21] also presented the general projective Riccati equations method to derive exact solutions for the combined sinh-cosh-Gordon equation by means of Painlevé property. Khalique and Magalakwe [22] derived another type of exact solutions of the combined sinh-cosh-Gordon equation by means of the Lie symmetry analysis along with the simplest equation method.…”

Section: Introductionmentioning

confidence: 99%

New exact solutions of the combined and double combined sinh-cosh-Gordon equations via modified Kudryashov method

Joardar¹,

Kumar²,

Woadud³

2018

IJPR

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The combined and double combined sinh-cosh-Gordon equations are very important to a wide range of various scientific applications that ranges from chemical reactions to water surface gravity waves. In this article, with the assistance of a function transform and Painlevè property, the nonlinear combined and double combined sinh-cosh-Gordon equations turn into ordinary differential equations. Later on, modified Kudryashov method is adopted for investigating new analytical solution of the studied equations. As a consequence, a series of new analytical solutions are acquired and we demonstrated the actual behavior of the achieved solutions of the mentioned equations with the aid of 3D and 2D MATLAB graphs. Finally, we also validate the effectiveness of the modified Kudryashov method for the problem of extracting new exact solutions of the combined and double combined sinh-cosh-Gordon equations with the aid of Maple package program. It is shown that the implemented method is capable to extract new solutions and it can also use to other nonlinear partial differential equation (NLPDE's) arising in mathematical physics or other applied field.

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Combined sinh-cosh-Gordon equation: Symmetry reductions, exact solutions and conservation laws

Cited by 18 publications

References 24 publications

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Symmetry analysis for three-dimensional dissipation Rossby waves

New exact solutions of the combined and double combined sinh-cosh-Gordon equations via modified Kudryashov method

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