The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

Zhao, Yun-Mei; He, Yinghui; Ye, Long

doi:10.1155/2013/960798

Cited by 15 publications

(10 citation statements)

References 24 publications

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“… Step 2. The considered modified version of the SEM (MSEM) expresses the solution of Equation by making an ansatz for v ( ξ ) as

v ((), ξ) = \sum_{i = 0}^{m} A_{i} {(\frac{ϕ^{'} ((), ξ)}{ϕ ((), ξ)})}^{i}, A_{m} \neq 0,

where A i ( i = 0,1, … ,m) are parameters to be determined, and m is a positive integer that can be determined by the balance algorithm as in Step 3 of SEM. ϕ ( ξ ) is an unspecified function to be determined subsequently. Step 3.…”

Section: The Simple Equation Methodsmentioning

confidence: 99%

“…The SEM and its variants have been successfully applied for finding exact solutions of nonlinear Schrödinger equation, coupled Konno‐Oono equations, nonlinear telegraph equation, generalized Davey‐Stewartson and Zakharov equations, dispersive water wave equations, generalized (2 + 1)‐dimensional nonlinear evolution equations (NLEEs), and the (4 + 1)‐dimensional Fokas equation . See also the included bibliography therein.…”

Section: Introductionmentioning

confidence: 99%

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New kink solutions for the van der Waals p‐system

Az-Zo’bi

2019

Math Methods in App Sciences

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Communicated by: Y. Qin MSC Classification: 35C07; 35M30; 35Q35; 47J35; 65Z05; 76Nxx The simple equation method and modified simple equation method are employed to seek exact traveling wave solutions to the (1 + 1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. Under the help of Mathematica, new classes of kink solutions are derived. Numerical simulations with special choices of the free parameters are displayed by threeand two-dimensional plots. The two methods demonstrate simplicity, reliability, and efficiency. KEYWORDS modified simple equation method, simple equation method, solitary waves, systems of mixed type, traveling wave solutions, van der Waals system

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“… Step 2. The considered modified version of the SEM (MSEM) expresses the solution of Equation by making an ansatz for v ( ξ ) as

v ((), ξ) = \sum_{i = 0}^{m} A_{i} {(\frac{ϕ^{'} ((), ξ)}{ϕ ((), ξ)})}^{i}, A_{m} \neq 0,

Section: The Simple Equation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New kink solutions for the van der Waals p‐system

Az-Zo’bi

2019

Math Methods in App Sciences

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“…We employ the simplest equation approach, as described in [55], to solve the second-order NLODE (2.12). The simplest equations used in this method are the Bernoulli and Riccati equations, given respectively by:…”

Section: Solutions Of Equation (13) Using the Simplest Equation Approachmentioning

confidence: 99%

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

Khalique,

Lephoko

2024

Commun. Theor. Phys.

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This paper is devoted to the investigation of the Landau-Ginzburg-Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov's method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in studying physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

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“…Reduction to ordinary differential equations is usually based on Lie symmetries and wave transformations. In literature recent years, lots of methods are given for solving NPDEs for example the tanh-coth strategy [1], the auxiliary equation technique [2,3], modified simple equation technique [4], Bernoulli function methodology [5], the new extended direct algebraic technique [6], the sine-Gordon expansion technique [7,8], Hirota bilinear technique [9], the simplest extended equation technique [10,11], the F-expansion technique [12], He's semi-inverse technique [13], the sub-ODE technique [14], the (G'/G) -expansion technique [15], the generalized Kudryashov technique [16], and many more. The common point of all the methods mentioned here is to convert the PDEs to the ordinary differential equations (ODEs) with the help of wave transformations.…”

Section: Introductionmentioning

confidence: 99%

Research on sensitivity analysis and traveling wave solutions of the (4 + 1)-dimensional nonlinear Fokas equation via three different techniques

Yalçın¹,

Akbulut²,

Raza³

2022

Phys. Scr.

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In the current manuscript, (4+1) dimensional Fokas nonlinear equation is considered to obtain traveling wave solutions. Three renowned analytical techniques, namely the generalized Kudryashov method (GKM), the modified extended tanh technique, exponential rational function method (ERFM) are applied to analyze the considered model. Distinct structures of solutions are successfully obtained. The graphical representation of the acquired results is displayed to demonstrate the behavior of dynamics of nonlinear Fokas equation. Finally, the proposed equation is subjected to a sensitive analysis.

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The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

Cited by 15 publications

References 24 publications

New kink solutions for the van der Waals p‐system

New kink solutions for the van der Waals p‐system

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

Research on sensitivity analysis and traveling wave solutions of the (4 + 1)-dimensional nonlinear Fokas equation via three different techniques

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