Derivation of soliton solutions to nonlinear evolution equations using He’s variational principle

Elboree, Mohammed K.

doi:10.1016/j.apm.2014.11.053

Cited by 22 publications

(13 citation statements)

References 24 publications

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“…The variational method proposed by professor Ji‐Huan He is a powerful method to seek the solitary wave solutions of many nonlinear partial differential equations 32–34 . In this section, we aim to get the solitary wave solution of Equation by He's variational method.…”

Section: Solitary Wave Solutionsmentioning

confidence: 99%

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

Wang

2021

Math Methods in App Sciences

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Yu‐Lan Ma, et al. (Mathematical Methods in the Applied Sciences, 2019,42(1)) make outstanding contributions for the soliton solutions of the generalized fourth‐order Boussinesq equation, which is used to describe the wave motion in fluid mechanics. But the periodic wave solution is not reported. So in this paper, He's variational methods are employed to find the solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation. The greatest advantage of the variational method is that it can reduce the order of the research equation, make the equation more simple, and obtain the optimal solution. Finally, the numerical results are shown in the form of a graph to prove the applicability and effectiveness of the method. The results reveal that variational method is simple and straightforward and can avoid the tedious calculation process, which is expected to shed a light to the new research frontiers of solitary wave theory.

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Section: Solitary Wave Solutionsmentioning

confidence: 99%

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

Wang

2021

Math Methods in App Sciences

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“…Step 5: Then taking the above assumed expressions into Equation (2.4), applying the Ritz-like method [24][25][26][27][28] with the stationary conditions, we have…”

Section: The Algorithm Of the Vdmmentioning

confidence: 99%

“…Inspired by He's variational theory, 24–28 we assume the solution of Equation () with the form as

ϕ ((), ξ) = M {sech}^{2} ((), ξ),

where M can be determined later. Now can take above hypothesis into Equation (), there is

\begin{matrix} lefttrue \\ J (M) = \int_{0}^{\infty} \{- \frac{1}{2} (m - {italicnv}^{2}) {([], - 2 M \sec h^{2} ((), ξ) \tanh ((), ξ))}^{2} + \frac{1}{2} (v^{2} - 1) {([], M \sec h^{2} ((), ξ))}^{2} - \frac{1}{2} v {([], M \sec h^{2} ((), ξ))}^{3}\} dξ \\ = - \frac{2}{15} M^{2} (5 + 4 m - 4 {italicnv}^{2} + 4 Mv - 5 v^{2}) . \end{matrix}

…”

Section: The Application Of the Vdmmentioning

confidence: 99%

Study on the explicit solutions of the Benney–Luke equation via the variational direct method

Wang

2021

Math Methods in App Sciences

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In this current work, we main to find the abundant explicit solutions of the Benney-Luke equation by a novel approach, which is called the variational direct method (VDM). The VDM based on the variational theory and Ritz-like method can reduce the order of the differential equation to make the equation simpler and obtain the optimal solution by the stationary conditions. The explicit solutions in terms of dark solitary, dark-like solitary, kinky-dark solitary, and periodic wave solutions are presented through the numerical results in the form of 3-D plot, 2-D curve, and 2-D contour. The results show that the VDM is simple, straightforward, and powerful, which is expected to open new perspectives for the traveling wave theory.

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“…Solitary waves or soliton solutions play an important role in nonlinear field, and to obtain soliton solutions is important in different areas of physics and technology. In recent years, analytical soliton solutions of some nonlinear differential equations have been gotten in various previous articles [10][11][12][13][14]16]. In this paper, we will employ the new proposed method to study the Vakhnenko equation [5][6][7][8]…”

Section: Introductionmentioning

confidence: 99%

New Analytical Approximations with Error Estimates for the One-loop Soliton Solution to the Vakhnenko Equation

Qin¹,

Zou²,

Wang³

et al. 2016

Proceedings of the 2nd International Conference on Electronics, Network and Computer Engineering (ICENCE 2016)

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In this paper, based on the traditional homotopy analysis method, we have successfully proposed a new analytical method, namely the piecewise perturbation method (PPM), for solving nonlinear problems. Here we introduce the idea of Newton iteration into the traditional homotopy analysis method to revise the initial guesses, which significantly increases the accuracy and the convergence rate of the series solutions. Further, we apply this method to obtain the one-loop soliton solution of the Vakhnenko equation in order to verify its potential and validity in solving nonlinear problems. With the aid of the optimal value of the convergence-control parameter determined by the averaged residual error technique, comparisons are made between the proposed method and the traditional homotopy analysis method. The results reveal that these new approximations with error estimates possess better accuracy and higher convergence rate than those obtained by the traditional homotopy analysis method.

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Derivation of soliton solutions to nonlinear evolution equations using He’s variational principle

Cited by 22 publications

References 24 publications

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

Study on the explicit solutions of the Benney–Luke equation via the variational direct method

New Analytical Approximations with Error Estimates for the One-loop Soliton Solution to the Vakhnenko Equation

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