Multiple-soliton solutions, soliton-type solutions and rational solutions for the $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional generalized shallow water equation in oceans, estuaries and impoundments

Zeng, Zhikai; Liu, Jian-Guo; Nie, Bin

doi:10.1007/s11071-016-2914-y

Cited by 36 publications

(8 citation statements)

References 30 publications

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“…Applying the painlevé analysis method, the Painlevé-Bäcklund transformation [43,44] of the (2+1)-dimensional dispersive long wave equation is written as follows…”

Section: Lump Solutionsmentioning

confidence: 99%

Rational solutions and their interaction solutions for the (2+1)-dimensional dispersive long wave equation

Zhang

2020

Phys. Scr.

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The Hirota bilinear form of the (2+1)-dimensional dispersive long wave equation by the truncated painlevé series in this paper is obtained. Meanwhile, a pair of quartic-linear forms are also constructed by an appropriate selection of seed solution to explore the lump solutions of the (2+1)-dimensional dispersive long wave equation. Then some novel interaction solutions by combining quadratic functions and exponential functions are yielded. Finally, in order to better illustrate the features of the results, we draw the three-dimensional and two-dimensional figures.

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“…Applying the painlevé analysis method, the Painlevé-Bäcklund transformation [43,44] of the (2+1)-dimensional dispersive long wave equation is written as follows…”

Section: Lump Solutionsmentioning

confidence: 99%

Rational solutions and their interaction solutions for the (2+1)-dimensional dispersive long wave equation

Zhang

2020

Phys. Scr.

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“…also called the second equation in the Kadomtsev-Petviashvili hierarchy, has been considered as a model to describe the propagation of the water wave in the ocean or weather simulations [19], where u is the wave-amplitude function of the scaled spatial coordinates x, y, z and temporal coordinate t. Some properties for equation (1) have been investigated [12,18,19]. Soliton solutions have been obtained via the hyperbolic functions [18] and Hirota bilinear method [19].…”

Section: Xy XX Y Xzmentioning

confidence: 99%

Pfaffian and extended Pfaffian solutions for a (3+1)-dimensional generalized wave equation

Lan

2019

Phys. Scr.

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Under investigation in this paper is a (3+1)-dimensional generalized wave equation, which is the second equation in the Kadomtsev–Petviashvili hierarchy. Linear partial differential conditions presented in this paper are different from the previous study. Pfaffian and extended Pfaffian solutions with a real function φ(y) have been obtained based on these linear partial differential conditions, where y is the scaled space coordinate, and compared with the solutions obtained in the previous articles, the advantage of our ones is that φ(y) could represent an inhomogeneous medium to influence or control the shape of waves. Specific solutions, namely the first- and second-order exact solutions from the Pfaffian and extended Pfaffian solutions are discussed. Discussions indicate that the first- and second-order exact solutions are of the same shape as the corresponding one- and two-soliton solutions on the x−t or z−t plane but of the different shape on the y−t plane.

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“…The interaction solutions of nonlinear partial differential equations are a topic of general interest in nonlinear systems. [1][2][3][4] Among them, shallow water wave equation has been one of the hottest issues in recent years, [5][6][7][8][9][10][11] such as marine engineering, hydrodynamics, mathematical physics in other fields. Because its exact solution is a special solution existing stably in space, [12] it has very important practical significance for many complex physical phenomena [13] and some nonlinear engineering problems.…”

Section: Introductionmentioning

confidence: 99%

“…( 2) by using the Hriota bilinear method, Darboux transformation method, etc. [6][7][8][9][10][11] Equation ( 1) can be transformed into the following form by using the Hriota bilinear method:…”

Section: Introductionmentioning

confidence: 99%

Fusionable and fissionable waves of (2+1)-dimensional shallow water wave equation

Wang¹,

Ding²,

Li³

2022

Chinese Phys. B

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We investigate a (2+1)-dimensional shallow water wave equation and describe its nonlinear dynamical behaviors in physics. Based on the N-soliton solutions, the higher-order fissionable and fusionable waves, fissionable or fusionable waves mixed with soliton molecular and breather waves can be obtained by various constraints of special parameters. At the same time, by the long wave limit method, the interaction waves between fissionable or fusionable waves with higher-order lumps are acquired. Combined with the dynamic figures of the waves, the properties of the solution are deeply studied to reveal the physical significance of the waves.

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Multiple-soliton solutions, soliton-type solutions and rational solutions for the $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional generalized shallow water equation in oceans, estuaries and impoundments

Cited by 36 publications

References 30 publications

Rational solutions and their interaction solutions for the (2+1)-dimensional dispersive long wave equation

Rational solutions and their interaction solutions for the (2+1)-dimensional dispersive long wave equation

Pfaffian and extended Pfaffian solutions for a (3+1)-dimensional generalized wave equation

Fusionable and fissionable waves of (2+1)-dimensional shallow water wave equation

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