Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water

Lü, Xing; Wang, Jianping; Lin, Fuhong; Zhou, Xianwei

doi:10.1007/s11071-017-3942-y

Cited by 87 publications

(20 citation statements)

References 46 publications

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“…Remark 1. When choosing N = 2, n i = 1 in expression (4), the rational solution is reduced to the lump solution [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][45][46][47][48][49][50][51][52][53][54][55][56][57].…”

Section: Rational Solution and Their Interaction Solutionmentioning

confidence: 99%

Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

Wang

Bilige

Pang

2020

Advances in Mathematical Physics

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In this paper, we gave a form of rational solution and their interaction solution to a nonlinear evolution equation. The rational solution contained lump solution, general lump solution, high-order lump solution, lump-type solution, etc. Their interaction solution contained the classical interaction solution, such as the lump-kink solution and the lump-soliton solution. As the example, by using the generalized bilinear method and symbolic computation Maple, we obtained abundant high-order lump-type solutions and their interaction solutions between lumps and other function solutions under certain constraints of the (3+1)-dimensional Jimbo-Miwa equation. Via three-dimensional plots, contour plots and density plots with the help of Maple, the physical characteristics and structures of these waves are described very well. These solutions have greatly enriched the exact solutions of the (3+1)-dimensional Jimbo-Miwa equation on the existing literature.

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Section: Rational Solution and Their Interaction Solutionmentioning

confidence: 99%

Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

Wang

Bilige

Pang

2020

Advances in Mathematical Physics

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“…For instance, the Hirota's bilinear transformation, the generalized bilinear transformation, the inverse‐scattering transformation, the Painlevé analysis approach, and the Darboux transformation . Recently, the lump‐type and mixed‐lump‐type solutions, analytical and rationally localized, of NPDEs have been extensively studied by many researchers . However, there is no systematic study on the lump solutions of any differential‐difference equations, such as the Toda equation.…”

Section: Introductionmentioning

confidence: 99%

Lump solutions of the 2D Toda equation

Sun

2020

Math Methods in App Sciences

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In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method.

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“…A tsunami wave is often characterized as a shallow-water wave since its wavelength is usually far longer than the water depth. Many mathematical models have been developed to describe the shallow water waves in which the typical models include the Korteweg-de Vries (KdV) equation [1] , the Boussinesq equation [2] , the Degasperis-Procesi equation [3] , the Benjamin-Bona-Mahony (BBM) equation [4] , the Kadomtsev-Petviashvili (K-P) equation [5] , and the Whitham-Broer-Kaup (WBK) model [6] . Particularly, the WBK model is usually used to describe the tsunami wave dynamics under gravity, which is formulated based on the assumption that the fluid is incompressible and irrotational.…”

Section: Introductionmentioning

confidence: 99%

New groups of solutions to the Whitham-Broer-Kaup equation

Wang

Sun

2020

Appl. Math. Mech.-Engl. Ed.

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The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.

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Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water

Cited by 87 publications

References 46 publications

Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

Lump solutions of the 2D Toda equation

New groups of solutions to the Whitham-Broer-Kaup equation

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