Traveling wave behavior of new (2+1)-dimensional combined KdV–mKdV equation

Vivas-Cortez, Miguel; Akram, Ghazala; Sadaf, Maasoomah; Arshed, Saima; Rehan, Kashif; Farooq, Kainat

doi:10.1016/j.rinp.2023.106244

Cited by 17 publications

(7 citation statements)

References 17 publications

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“…The EFM [16] proposed by Ji-Huan He and Xu-Hong Wu in 2006 provides us with a straightforward and effective method for obtaining generalized solitary wave solutions and periodic solutions of NLEEs. The method has been applied to many kinds of equations like the double sine-Gordon equation [17], Burger equations [18], Maccari's system [19], the Klein-Gordon equation [20], the combined KdV-mKdV equation [21], variant Boussinesq equations [22], the Broer-Kaup-Kupershmidt equations [23], variable-coefficient equations [24], high-dimensional equations, discrete equations and so on [25][26][27]. In 2009, Dai et al [28] generalized the EFM to solve stochastic equations.…”

Section: Introductionmentioning

confidence: 99%

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Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

O’Regan,

Aderyani,

Saadati

et al. 2024

Axioms

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This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method.

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Section: Introductionmentioning

confidence: 99%

“…and putting our obtained results into Equation (21), we obtain the generalized solution of Equation ( 15) as follows:…”

mentioning

confidence: 99%

Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

O’Regan,

Aderyani,

Saadati

et al. 2024

Axioms

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“…Numerous strategies for obtaining various soliton solutions to these NLPDEs have been discovered in the literature. Among these are the square operator approach (Wang et al 2022), the unified technique (Vivas-Cortez et al 2023), Hirota's bilinear approach , the neural network approach (Xu et al 2023), the extended tanh-function approach (Zaman et al 2023a, b), and many others (Arefin et al 2022;. They have enhanced the dynamic approach by employing various methods, such as the generalized Kudryashov method (Akbar et al 2022), the sinh-Gordon equation expansion method (Akbar et al 2021), the (G � ∕G)-expansion , the tanh technique (Almusawa et al 2022), the rational (1∕Φ � )-expansion approach ).…”

Section: Introductionmentioning

confidence: 99%

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

Ansari,

Jhangeer,

Imran

et al. 2024

Opt Quant Electron

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This manuscript delves into the examination of the stochastic fractional derivative of Drinfel’d-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena.

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“…The exploration of a diverse range of nonlinear partial differential equations (NLPDEs) is necessary for mathematically modeling complex phenomena that evolve over time. Researchers have developed various methodologies, including numerical and analytical techniques, as well as computational algorithms, to effectively solve PDEs and gain insights into complex phenomena [5]. Some of these techniques include, Homotopy analysis [6], extended generalized Kudryashov method [7], Laplace-Adomian decomposition method [8], Riccati projective equation method [9], Hirota bilinear form [10], Ba ¨cklund transformation [11], and many others [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics

Vivas-Cortez,

Rani,

Raza

et al. 2024

PLoS ONE

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This research paper focuses on the study of the (3+1)-dimensional negative order KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, an important nonlinear partial differential equation in oceanography. The primary objective is to explore various solution techniques and analyze their graphical representations. Initially, two wave, three wave, and multi-wave solutions of the negative order KdV CBS equation are derived using its bilinear form. This analysis shed light on the behavior and characteristics of the equation’s wave solutions. Furthermore, a bilinear Bäcklund transform is employed by utilizing the Hirota bilinear form. This transformation yields exponential and rational function solutions, contributing to a more comprehensive understanding of the equation. The resulting solutions are accompanied by graphical representations, providing visual insights into their structures. Moreover, the extended transformed rational function method is applied to obtain complexiton solutions. This approach, executed through the bilinear form, facilitated the discovery of additional solutions with intriguing properties. The graphical representations, spanning 2D, 3D, and contour plots, serve as valuable visual aids for understanding the complex dynamics and behaviors exhibited by the equation’s solutions.

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Traveling wave behavior of new (2+1)-dimensional combined KdV–mKdV equation

Cited by 17 publications

References 17 publications

Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics

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