Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation

Tian, Shou‐Fu

doi:10.1016/j.aml.2019.106056

Cited by 137 publications

(42 citation statements)

References 20 publications

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“…Equation changes into the following well‐known fourth‐order nonlinear Boussinesq water wave equation when m = 1, n = 1, and k = − 1 as follows:

η_{tt} - η_{xx} - {(η^{2})}_{xx} - η_{italicxx x x} = 0,

which arises in propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in a plasma and one‐dimensional nonlinear lattice‐waves, etc. There have been many soliton solutions available for Equation 17–20 . However its periodic wave solution is are rarely reported.…”

Section: Introductionmentioning

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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“…Equation changes into the following well‐known fourth‐order nonlinear Boussinesq water wave equation when m = 1, n = 1, and k = − 1 as follows:

η_{tt} - η_{xx} - {(η^{2})}_{xx} - η_{italicxx x x} = 0,

Section: Introductionmentioning

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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“…Since the success rate of NLEEs is high in illustrating versatile problems in different sectors, searching solitary wave solutions has gained popularity to the researchers. Thus, a number of methods have been developed by various researchers to carry out exact and explicit stable soliton solutions of nonlinear physical models, such as, the tanh-function expansion and its various modifications [ 4 ], the exp-function method [ 5 ], the ansatz method [ 6 ], the sine-cosine method [ 7 ], the F-expansion method [ 8 ], the complex hyperbolic-function method [ 9 ], the variational iteration method [ 10 ], the

-expansion method [ 11 ], the Jacobi elliptic function method [ 12 ], the improved Bernoulli sub-equation function method [ 13 ], the homotopy analysis method [ 14 ], the Adomian decomposition method [ 15 ], the modified extended tanh method [ 16 ], the

-expansion method [ 17 ], the finite element method [ 18 ], the first integral method [ 19 ], the alternative

expansion method [ 20 ], the modified simple equation method [ 21 ], the modified two-component Dullin-Gottwald-Holm (mDGH2) system [ 22 ], the Riemann-Hilbert method [ 23 , 24 , 25 ], the Lie symmetry method [ 26 ], the long wave limit method [ 27 ], the truncated Painlevé expansion method [ 28 ], the sine-Gordon expansion (SGE) method [ 29 , 30 , 31 , 32 , 33 , 34 ] and several type of soliton [ 35 , 36 ] process. The reputed sine-Gordon equation method was developed based on the wave transformation and it functions only for lower-dimensional NLEEs.…”

Section: Introductionmentioning

confidence: 99%

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis

Kundu

Fahim

Islam

et al. 2021

Heliyon

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The Estevez-Mansfield-Clarkson (EMC) equation and the (2+1)-dimensional Riemann wave (RW) equation are important mathematical models in nonlinear science, engineering and mathematical physics which have remarkable applications in the field of plasma physics, fluid dynamics, optics, image processing etc. Generally, through the sine-Gordon expansion (SGE) method only the lower-dimensional nonlinear evolution equations (NLEEs) are examined. However, the method has not yet been extended of finding solutions to the higher-dimensional NLEEs. In this article, the SGE method has been developed to rummage the higher-dimensional NLEEs and established steady soliton solutions to the earlier stated NLEEs by putting in use the extended higher-dimensional sine-Gordon expansion method. Scores of soliton solutions are figure out which confirms the compatibility of the extended SGE method. The solutions are analyzed for both lower and higher-dimensional nonlinear evolution equations through sketching graphs for alternative values of the associated parameters. From the figures it is notable to perceive that the characteristic of the solutions depend upon the choice of the parameters. This study might play an impactful role in analyzing higher-dimensional NLEEs through the extended SGE approach.

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“…Methods based on the framework of symmetry and conservation law analysis can be systematically applied to wide classes of partial differential equation and ordinary differential equation models, which needs to be paid more attention by socio-economic scientists, computer engineers, mathematicians, etc [ 39 , 40 ]. The recent advances in nonlinear integrable systems may open a new route in the search for the integrability and analytical methods of high-order nonlinear differential equation, symmetric equations and discrete equations, including the transformation relationship between equations, the construction of integrable clusters, symmetry and conservation laws, soliton solutions and quasi-periodic wave solutions and integrable properties [ 41 – 45 ]. Inescapably, the nonlocal symmetry method for many nonlinear systems has been successfully studied by using the truncated Painlevé method and the Möbious (conformal) invariant form [ 46 , 47 ].…”

Section: Introductionmentioning

confidence: 99%

Lie symmetry analysis of the effects of urban infrastructures on residential property values

et al. 2021

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Due to the complexity of socio-economic-related issues, people thought of housing market as a chaotic nucleus situated at the intersection of neighboring sciences. It has been known that the dependence of house features on the residential property value can be estimated employing the well-established hedonic regression analysis method in teams of location characteristic, neighborhood characteristic and structure characteristic. However, to further assess the roles of urban infrastructures in housing markets, we proposed a new kind of volatility measure for house prices utilizing the Lie symmetry analysis of quantum theory based on Schrödinger equation, mainly focusing on the effects of transportation systems and public parks on residential property values. Based on the municipal open government data regularly collected for four cities, including Boston, Milwaukee, Taipei and Tokyo, and all spatial sampling sites were featured by United States Geological Survey (USGS) National Map, transportation and park were modelled as perturbations to the quantum states generated by the feature space in response to the environmental amenities with different spatial extents. In an attempt to ascertain the intrinsic impact of the location-dependent price information obtained, the similarity functions associated with the Schrödinger equation were considered to facilitate revealing the city amenities capitalizing into house prices. By examining the spatial spillover phenomena of house prices in the four cities investigated, it was found that the mass transit systems and the public green lands possessed the infinitesimal generators of Lie point symmetries Y2 and Y5, respectively. Compared statistically with the common performance criteria, including mean absolute error (MAE), mean squared error (MSE) and, root mean squared error (RMSE) obtained by hedonic pricing model, the Lie symmetry analysis of the Schrödinger equation approach developed herein was successfully carried out. The invariant-theoretical characterizations of economics-related phenomena are consonant with the observed residential property values of the cities internationally, ultimately leading to develop a new perspective in the global financial architecture.

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Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation

Cited by 137 publications

References 20 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

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis

Lie symmetry analysis of the effects of urban infrastructures on residential property values

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