A (2+1)-dimensional combined KdV–mKdV equation: integrability, stability analysis and soliton solutions

Malik, Sandeep; Kumar, Sachin; Das, Achintya

doi:10.1007/s11071-021-07075-x

Cited by 25 publications

(8 citation statements)

References 29 publications

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“…In this study, we use the exp(−ϕ(z))-expansion method to obtain abundant new exact solutions to the (2+1)-dimensional combined KdV-mKdV equation. Except the types of hyperbolic and exponential function solutions which are the same as those of Sandeep Malik's paper [25], we also get new types of function solutions including trigonometric and rational solutions. Additionally, the results indicate that utilizing the exp(−ϕ(z))-expansion method to the combined KdV-mKdV equation and the (2+1)-dimensional combined KdV-mKdV equation can get the same forms of solutions, while the solutions to the (2+1)-dimensional combined KdV-mKdV equation have one more case.…”

Section: Discussionmentioning

confidence: 98%

“…Then Malik et al [25] proposed the (2+1)-dimensional combined KdV-mKdV equation by combining them, which is given by:…”

Section: Introductionmentioning

confidence: 99%

“…(1.1) and (1.2) respectively. Additionally, although the authors have obtained the analytical solutions to the combined KdV-mKdV equation in [18], the (2+1)-dimensional combined KdV-mKdV equation in [25] has more dimensions and the mixed partial derivatives in contrast to the former, which means a much broader researching space for scholars. Therefore, it makes sense to research the the (2+1)-dimensional combined KdV-mKdV equation deeply.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

CHEN

2022

Fundamental Journal of Mathematics and Applications

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In this paper, we obtain exact solutions of the (2+1)-dimensional combined KdV-mKdV equation by using symbol calculation approach. First, we give some background on the equation. Second, the exp(-\varphi(z))-expansion method will be introduced to solve the equation. After, using the exp(-\varphi(z))-expansion method to solve the equation, we can get four types of exact solutions, which are hyperbolic, trigonometric, exponential, and rational function solutions. Finally, we can observe the characteristics of the exact solutions via computer simulation more easily.

show abstract

Section: Discussionmentioning

confidence: 98%

“…Then Malik et al [25] proposed the (2+1)-dimensional combined KdV-mKdV equation by combining them, which is given by:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

CHEN

2022

Fundamental Journal of Mathematics and Applications

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“…Therefore, nonlinear partial differential equations (NPDEs) play a major role in modeling of these natural waves and several wave phenomena as well. The massive applications of NPDEs can be seen in various fields of mathematical sciences, biological sciences, nonlinear optics, electromagnetic theory, quantum theory, optical fiber, plasma physics, heat transfer, fluid dynamics, and so forth [1‐32,33,34]. The NPDEs are difficult to handle with traditional methods, so a large number of methods such as Darboux transformations [1], Fan's subequation method [3], extended mapping method [4], sub‐ODE method [5], extended tanh method [6], amplitude ansatz method [7], WTC truncation method [8], nonlocal symmetry method [9], and similarity transformation method [10–34] are evolved to derive exact solutions.…”

Section: Introductionmentioning

confidence: 99%

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

Kumar,

Anand,

Tanwar

2024

Math Methods in App Sciences

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The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

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“…A detailed study of the combined KdV-mKdV equation in (2+1) dimensions has recently been presented, investigating its integrability, stability analysis, and solitonic solutions [23]. In addition, analytical solutions have been obtained using methods such as those mentioned above [31].…”

Section: Introductionmentioning

confidence: 99%

New analytical solutions to the stochastic (2 + 1)-dimensional combined KdV-mKdV equation

Rosas

Saavedra

Morales

2023

Preprint

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In this article, we investigate the (2 + 1)-dimensional stochastic KdV-mKdv equation perturbed by white noise in the Itô sense. The improved Tanh-Coth method is used to obtain new stochastic solutions. It also extends some previous research. Since this new equation is essential for describing various phenomena in ocean engineering, hydrodynamics and optics, the obtained solutions are useful for interpreting certain interesting physical phenomena. To show how multiplicative white noise affects the exact solution, we plot the solutions in Julia and show some 3D plots.

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A (2+1)-dimensional combined KdV–mKdV equation: integrability, stability analysis and soliton solutions

Cited by 25 publications

References 29 publications

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

New analytical solutions to the stochastic (2 + 1)-dimensional combined KdV-mKdV equation

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